Fusion based quantum computing

ABSTRACT

A method includes receiving a plurality of quantum systems, wherein each quantum system of the plurality of quantum system includes a plurality of quantum sub-systems in an entangled state, and wherein respective quantum systems of the plurality of quantum systems are independent quantum systems that are not entangled with one another. The method further includes performing a plurality of joint measurements on different quantum sub-systems from respective ones of the plurality of quantum systems, wherein the joint measurements generate joint measurement outcome data and determining, by a decoder, a plurality of syndrome graph values based on the joint measurement outcome data.

CROSS-REFERENCE TO RELATED APPLICATION

This patent application claims the benefit of U.S. Provisional PatentApplication No. 62/967,513, filed Jan. 29, 2020, and also claims thebenefit of U.S. Provisional patent Application No. 63/140,210, filedJan. 21, 2021. The disclosures of both applications are herebyincorporated by reference in their entirety for all purposes.

TECHNICAL FIELD

One or more embodiments of the present disclosure relate generally toquantum computing devices and methods and, more specifically, to faulttolerant quantum computing devices and methods.

BACKGROUND

In fault tolerant quantum computing, quantum error correction isrequired to avoid an accumulation of qubit errors that then leads toerroneous computational outcomes. One method of achieving faulttolerance is to employ error correcting codes (e.g., topological codes)for quantum error correction. More specifically, a collection ofphysical qubits can be generated in an entangled state (also referred toherein as an error correcting code) that encodes for a single logicalqubit that is protected from errors.

In some quantum computing systems, cluster states of multiple qubits,or, more generally, graph states can be used as the error correctingcode. A graph state is a highly entangled multi-qubit state that can berepresented visually as a graph with nodes representing qubits and edgesrepresenting entanglement between the qubits. However, various problemsthat either inhibit the generation of entangled states or destroy theentanglement once created have frustrated advancements in quantumtechnologies that rely on the use of highly entangled quantum states.

Furthermore, in some qubit architectures, e.g., photonic architectures,the generation of entangled states of multiple qubits is an inherentlyprobabilistic process that may have a low probability of success.

Accordingly, there remains a need for improved systems and methods forquantum computing that do not necessary rely on large cluster states ofqubits.

SUMMARY

Described herein are embodiments of fault-tolerant systems and methodsfor quantum computing that do not necessary rely on large cluster statesof qubits.

According to some embodiments, a method can comprise: receiving aplurality of quantum systems, wherein each quantum system of theplurality of quantum system includes a plurality of quantum sub-systemsin an entangled state, and wherein respective quantum systems of theplurality of quantum systems are independent quantum systems that arenot entangled with one another; performing a plurality of destructivejoint measurements (such as fusion operations) on different quantumsub-systems from respective ones of the plurality of quantum systems,wherein the destructive joint measurements destroy the different quantumsub-systems and generate joint measurement outcome data and transferquantum state information from the different quantum sub-systems toother unmeasured quantum sub-systems from the plurality of quantumsystems; and determining a logical qubit state based on the jointmeasurement outcome data. The logical qubit state can be determined in afault tolerant manner.

According to some embodiments, a method can comprise: receiving aplurality of quantum systems, wherein each quantum system of theplurality of quantum system includes a plurality of quantum sub-systemsin an entangled state, and wherein respective quantum systems of theplurality of quantum systems are independent quantum systems that arenot entangled with one another; performing a logical qubit gate byperforming a plurality of destructive joint measurements (such as fusionoperations) on different quantum sub-systems from respective ones of theplurality of quantum systems, wherein the destructive joint measurementsdestroy the different quantum sub-systems and generate joint measurementoutcome data and transfer quantum state information from the differentquantum sub-systems to other unmeasured quantum sub-systems from theplurality of quantum systems; and determining a result of the logicalqubit gate based on the joint measurement outcome data. The result ofthe logical qubit gate can be determined in a fault tolerant manner.

According to some embodiments, a quantum computing apparatus cancomprise: a qubit entangling system to generate a plurality of quantumsystems, wherein each quantum system of the plurality of quantum systemsincludes a plurality of quantum sub-systems in an entangled state, andwherein respective quantum systems of the plurality of quantum systemsare independent quantum systems that are not entangled with one another;a qubit fusion system to perform a plurality of destructive jointmeasurements on different quantum sub-systems from respective ones ofthe plurality of quantum systems, wherein the destructive jointmeasurements destroy the different quantum sub-systems and generatejoint measurement outcome data and transfer quantum state informationfrom the different quantum sub-systems to other unmeasured quantumsub-systems from the plurality of quantum systems; and a classicalcomputing system to determine a logical qubit state based on the jointmeasurement outcome data.

According to some embodiments, a quantum computing apparatus cancomprise: a qubit entangling system to generate a plurality of quantumsystems, wherein each quantum system of the plurality of quantum systemsincludes a plurality of quantum sub-systems in an entangled state, andwherein respective quantum systems of the plurality of quantum systemsare independent quantum systems that are not entangled with one another;a qubit fusion system to perform a logical qubit gate by performing aplurality of destructive joint measurements on different quantumsub-systems from respective ones of the plurality of quantum systems,wherein the destructive joint measurements destroy the different quantumsub-systems and generate joint measurement outcome data and transferquantum state information from the different quantum sub-systems toother unmeasured quantum sub-systems from the plurality of quantumsystems; and a classical computing system to determine a result of thelogical qubit gate based on the joint measurement outcome data.

According to some embodiments a method includes receiving, by a qubitfusion system, a plurality of quantum systems, wherein each quantumsystem of the plurality of quantum system includes a plurality ofquantum sub-systems in an entangled state. Respective quantum systems ofthe plurality of quantum systems are independent quantum systems thatare not entangled with one another. The method further includesperforming, by the qubit fusion system, a plurality of jointmeasurements on different quantum sub-systems from respective ones ofthe plurality of quantum systems. The joint measurements generate jointmeasurement outcome data. The method further includes determining, by adecoder, a plurality of syndrome graph values based on the jointmeasurement outcome data.

According to some embodiments performing the joint measurements includesperforming fusion operations.

According to some embodiments performing the joint measurements includeperforming a destructive joint measurement via a Type II fusionoperation.

According to some embodiments performing the plurality of jointmeasurements on different quantum sub-systems from respective ones ofthe plurality of quantum systems includes performing the plurality ofjoint measurements on only a subset of the plurality of quantumsub-systems that are received by the qubit fusion system therebyresulting in a subset of unmeasured quantum sub-systems.

According to some embodiments, the method further includes, receiving,by the qubit fusion system, a second plurality of quantum systems,wherein each quantum system of the second plurality of quantum systemincludes a second plurality of quantum sub-systems in an entangledstate, and wherein respective quantum systems of the second plurality ofquantum systems are independent quantum systems that are not entangledwith one another. The method further includes receiving the subset ofunmeasured quantum sub-systems and performing, by the qubit fusionsystem, a second plurality of joint measurements between i) secondquantum sub-systems from respective ones of the plurality of secondquantum systems and ii) respective quantum sub-systems from the subsetof unmeasured quantum sub-systems. The second plurality of jointmeasurements generates second joint measurement outcome data.

According to some embodiments a system includes a qubit fusion systemcomprising a plurality of fusion gates. The qubit fusion system isconfigured to receive a plurality of quantum systems, wherein eachquantum system of the plurality of quantum system includes a pluralityof quantum sub-systems in an entangled state, and wherein respectivequantum systems of the plurality of quantum systems are independentquantum systems that are not entangled with one another.

According to some embodiments the plurality of fusion gates are eachconfigured to perform a joint measurement on different quantumsub-systems from respective ones of the plurality of quantum systems,wherein the joint measurements generate joint measurement outcome data.

The system further includes a decoder communicatively coupled to thequbit fusion system and configured to receive the joint measurementoutcome data and to determine a plurality of syndrome graph values basedon the joint measurement outcome data.

According to some embodiments the fusion gates include a photoniccircuit and the plurality of quantum systems comprise photons as thequantum sub-systems, wherein the photonic circuit comprises a Type IIfusion gate.

According to some embodiments the joint measurement comprises atwo-particle projective measurement onto a Bell basis.

According to some embodiments the system further includes a quantummemory, coupled to at least one the qubit fusion system and that receiveand store a subset of the plurality of quantum sub-systems.

According to some embodiments the quantum memory is an optical fiber.

According to some embodiments the quantum memory is coupled to the qubitfusion system such that the joint measurement is performed between i)quantum sub-systems from respective ones of the plurality of quantumsystems and ii) respective quantum sub-systems from the subset of theplurality of quantum sub-systems that are stored in the quantum memory.

According to some embodiments the system of further includes a qubitentangling system that is configured to generate the plurality ofquantum systems.

According to some embodiments the qubit entangling system includes aquantum gate array.

According to some embodiments the qubit entangling system includes aphoton source system that is optically connected to an entangled stategenerator.

According to some embodiments the entangled state generator isconfigured to receive output photons from the photon source system andconvert the output photons to an entangled photonic state.

According to some embodiments the qubit entangling system includes aplurality of output waveguides that are optically coupled to the qubitfusion system and are configured to provide the entangled photonic stateto inputs of the fusion gates.

The following detailed description, together with the accompanyingdrawings, will provide a better understanding of the nature andadvantages of the claimed invention.

BRIEF DESCRIPTION OF THE DRAWINGS

Aspects of the present disclosure are illustrated by way of example.Non-limiting and non-exhaustive aspects are described with reference tothe following figures, wherein like reference numerals refer to likeparts throughout the various figures unless otherwise specified.

FIGS. 1A-1C are diagrams illustrating a cluster state and correspondingsyndrome graph for an entangled state of physical qubits in accordancewith some embodiments.

FIG. 2 shows a quantum computing system in accordance with one or moreembodiments.

FIG. 3 shows a quantum computing system in accordance with someembodiments.

FIG. 4 illustrates an example of a qubit entangling system in accordancewith some embodiments.

FIG. 5 shows one example of qubit fusion system in accordance with someembodiments.

FIG. 6 shows one possible example of a fusion site as configured tooperate with a fusion controller to provide measurement outcomes to adecoder for fault tolerant quantum computation in accordance with someembodiments.

FIGS. 7A-7C illustrates a fusion based quantum computing scheme forfault tolerant quantum computation in accordance with one or moreembodiments.

FIGS. 8A-8C show one example of a lattice preparation protocol forfusion based quantum computing in accordance with some embodiments.

FIGS. 9A-9B show one example of a lattice preparation protocol forfusion based quantum computing in accordance with some embodiments.

FIGS. 10A-10E shows a flow chart and example lattice preparationprotocol for illustrating a method for fusion based quantum computing inaccordance with one or more embodiments.

FIGS. 11A-11E show representations of dual-rail-encoded photonic qubitsand photonic circuits for performing unitary operation on photonicqubits in accordance with some embodiments.

FIGS. 12A-12B show representations of dual-rail-encoded photonic qubitsand photonic circuits for performing unitary operation on photonicqubits in accordance with some embodiments.

FIG. 13 shows photonic implementations of beam splitters that may beused to implement one or more spreaders, e.g., Hadamard gates, accordingto some embodiments.

FIG. 14 shows photonic implementations of beam splitters that may beused to implement one or more spreaders, e.g., Hadamard gates, accordingto some embodiments.

FIG. 15 shows one example of a Bell state generator circuit that can beused in some dual-rail-encoded photonic embodiments.

FIG. 16 shows an example of a Type II fusion circuit for a polarizationencoding according to some embodiments.

FIG. 17 shows an example of a Type II fusion circuit for a path encodingaccording to some embodiments.

FIGS. 18A-18D show effects of fusion in the generation of a clusterstate according to some embodiments.

FIG. 19 shows examples of Type II fusion gates boosted once inpolarization and path encodings according to some embodiments.

FIG. 20 shows a table with variations of the Type II fusion gate fordifferent measurement basis in a polarization encoding.

FIG. 21 shows examples of photonic circuit variations of the Type IIfusion gate for different choice of measurement basis in a path encodingaccording to some embodiments.

DETAILED DESCRIPTION

Reference will now be made in detail to embodiments, examples of whichare illustrated in the accompanying drawings. In the following detaileddescription, numerous specific details are set forth in order to providea thorough understanding of the various described embodiments. However,it will be apparent to one of ordinary skill in the art that the variousdescribed embodiments may be practiced without these specific details.In other instances, well-known methods, procedures, components,circuits, and networks have not been described in detail so as not tounnecessarily obscure aspects of the embodiments.

1. INTRODUCTION TO QUANTUM COMPUTING

Quantum computation is often considered in the framework of ‘CircuitBased Quantum Computation’ (CBQC) in which operations (or gates) areperformed on physical qubits. Gates can be either single qubit unitaryoperations (rotations), two qubit entangling operations such as the CNOTgate, or other multi-qubit gates such as the Toffoli gate.

Measurement Based Quantum Computation (MBQC) is another approach toimplementing quantum computation. In the MBQC approach, computationproceeds by first preparing a particular entangled state of many qubits,commonly referred to as a cluster state, and then carrying out a seriesof single qubit measurements on the cluster state to enact the quantumcomputation. In this approach, the choice of single qubit measurementsis dictated by the quantum algorithm being run on the quantum computer.In the MBQC approach, fault tolerance can be achieved by careful designof the cluster state and using the topology of this cluster state toencode logical qubits that is protected against any logical errors thatmay be caused by errors on any of the physical qubits that make up thecluster state. In practice, the value of the logical qubit can bedetermined, i.e., read out, based on the results (also referred toherein as measurement outcomes) of the single-particle measurements thatare made on the cluster state's physical qubits as the computationproceeds.

However, the generation and maintenance of long-range entanglementacross the cluster state and subsequent storage of large cluster statescan be a challenge. For example, for any physical implementation of theMBQC approach, a cluster state containing many thousands, or more, ofmutually entangled qubits must be prepared and then stored for someperiod of time before the single-qubit measurements are performed. Forexample, to generate a cluster state representing a single logical errorcorrected qubit, each of the collection of underlying physical qubitscan be prepared in the |+

state and a controlled-phase gate (CZ) state can be applied between eachphysical qubit pair to generate the overall cluster state. Moreexplicitly, a cluster state of highly entangled qubits can be describedby the undirected graph G=(V, E) with V and E denoting the sets ofvertices and edges, respectively and can be generated as follows: 1)initialize all the physical qubits to be in the |+

state, where

${ | +  \rangle = \frac{  { ( | 0   \rangle +} \middle| 1 \rangle )}{\sqrt{2}}};$

and 2) apply the controlled-phase gate (CZ) to each pair i,j of qubits.Accordingly, any cluster state, which physically corresponds to a largeentangled state of physical qubits, can be described as

|Ψ

_(graph)=Π_((i,j)∈E) CZ _(i,j)|+

^(⊗|V|)  (1)

where the CZ_(i,j) is the controlled phase gate operator and with V andE as defined above. Graphically, the cluster states defined by Eq. (1)can also be represented by a graph with vertices V that represent thephysical qubits (initialized in the |+

state) and edges E that represent entanglement between them (i.e., theapplication of the various CZ gates). In some cases, e.g., casesinvolving a fault tolerant MBQC scheme, |Ψ

_(graph) graph can take the form of graph in 3 dimensions. Like theexamples shown in FIG. 1A and FIG. 7C, such a graph can have a regularstructure formed from repeating unit cells and is therefore oftenreferred to as a “lattice.” When represented as a 3-dimensional lattice,2-dimensional boundaries of this lattice can be identified. Qubitsbelonging to those boundaries are referred to as “boundary qubits” whileall other qubits are referred to as “bulk qubits”.

After |Ψ

_(graph) is generated, this large state of mutually entangled qubitsmust be preserved long enough for a stabilizer measurement to beperformed, e.g., by making X measurements on all physical qubits in thebulk of the lattice and Z measurements on the boundary qubits.

FIG. 1A shows one example of a fault tolerant cluster state that can beused in MBQC, the topological cluster state introduced by Raussendorf etal., and commonly referred to as the Raussendorf Lattice as described infurther detail in Robert Raussendorf, Jim Harrington, and Kovid Goyal.A., Fault-Tolerant One-Way Quantum Computer, Annals of Physics,321(9):2242-2270, 2006. The cluster state is in the form of repeatinglattice cells (e.g., cell 120) with physical qubits (e.g., physicalqubit 116) arranged on the faces and edges of the cells. Entanglementbetween the physical qubits is represented by edges that connect thephysical qubits (e.g., edge 118), with each edge representing theapplication of the CZ gate, as described above in reference to Eq. (1).The cluster state shown here is merely one example among many and othertopological error correcting codes can be used without departing fromthe scope of the present disclosure. For example, volume codes such asthose disclosed within International Patent Application Publication No.WO/2019/173651, the contents of which is hereby incorporated byreference in its entirety for all purposes, can be used. Also the codesbased on non-cubical unit cells described in International PatentApplication Publication No. WO/2019/178009, the contents of which ishereby incorporated by reference in its entirety for all purposes, canbe used without departing from the scope of the present disclosure.Furthermore, while the example shown here is represented in threespatial dimensions, the same structure may also be obtained from otherimplementations of codes that are not based on a purely spatialentangled cluster state, but rather can include both entanglement in 2Dspace and entanglement in time, e.g., a 2+1D surface code implementationcan be used or any other foliated code. For cluster state implementationof such codes, all of the quantum gates needed for fault tolerantquantum computation can be constructed by making a series of singleparticle measurements to the physical qubits that make up the lattice.

Returning to FIG. 1A, a chunk of a Raussendorf lattice is shown. Such anentangled state can be used to encode one or more logical qubits (i.e.,one or more error corrected qubits) using many entangled physicalqubits. The collection of single particle measurement results of themultiple physical qubits (e.g., physical qubit 116) can be used forcorrecting errors and for performing fault tolerant computations on thelogical qubits through the use of a decoder. Many decoders are availablewith one example being the Union-Find decoder as described inInternational Patent Application Publication No. WO2019/002934A1, thedisclosure of which is hereby incorporated by reference in its entiretyfor all purposes. One of ordinary skill will appreciate that the numberof physical qubits required to encode a single logical qubit can varydepending on the precise nature of the physical errors, noise, etc.,that are experienced by the physical qubits, but to achieve faulttolerance, all proposals to date require entangled states of thousandsof physical qubits to encode a single logical qubit. Generating andmaintaining such a large entangled state remains a key challenge for anypractical implementation of the MBQC approach.

FIGS. 1B-1C illustrate how the decoding of a logical qubit can proceedfor a cluster state based on the Raussendorf lattice. As can be seen inFIG. 1A, the geometry of the cluster state is related to the geometry ofa cubic lattice (lattice cell 120) shown superimposed on the clustersstate in FIG. 1A. FIG. 1B shows the single particle measurement results(also superimposed on the cubic lattice) after the state of eachphysical qubit of the cluster state has been measured, with themeasurement results being placed in the former position of the physicalqubit that was measured (for clarity only measurement results thatresult from measurements of the surface qubits are shown).

In some embodiments, a measured qubit state can be represented by anumerical bit value of either 1 or 0 after all qubits have beenmeasured, e.g., in the x basis, with the 1 bit value corresponding tothe +x measurement outcome and the 0 but value corresponding to −xmeasurement outcomes (or vice versa). There are two types of qubits,those that are located on the edges of a unit cell (e.g. at edge qubit122), and those that are located on the faces of a unit cell (e.g., facequbit 124). In some cases, a measurement of the qubit may not beobtained, or the result of the qubit measurement may be invalid. Inthese cases, there is no bit value assigned to the location of thecorresponding measured qubit, but instead the outcome is referred toherein as an erasure, illustrated here as thick line 126, for example.These measurement outcomes that are known to be missing can bereconstructed during the decoding procedure.

To identify errors in the physical qubits, a syndrome graph can begenerated from the collection of measurement outcomes resulting from themeasurements of the physical qubits. For example, the bit valuesassociated with each edge qubit can be combined to create a syndromevalue associated with the vertex the results from the intersection ofthe respective edges, e.g., vertex 128 as shown in FIG. 1B. A set ofsyndrome values, also referred to herein as parity checks, areassociated with each vertex of the syndrome graph, as shown in FIG. 1C.More specifically, in FIG. 1C, the computed values of some of the vertexparity checks of the syndrome graph are shown. In some embodiments, aparity computation entails determining whether the sum of the edgevalues incident on a given vertex is an even or odd integer, with theparity result for that vertex being defined to be the result of the summod 2. If no errors occurred in the quantum state, or in the qubitmeasurement then all syndrome values should be even (or 0). On thecontrary, if an error occurs, it will result in some odd (or 1) syndromevalues. Only half of the bit values from qubit measurement areassociated with the syndrome graph shown (the bits aligned with theedges of the syndrome graph). There is another syndrome graph thatcontains all the bit values associated with the faces of the latticeshown. This leads to an equivalent decoding problem on these bits.

As mentioned above, the generation and subsequent storage of largecluster states of qubits can be a challenge. However, some embodiments,methods and systems described herein provide for the generation of a setof classical measurement data (e.g., a set of classical datacorresponding to syndrome graph values of a syndrome graph) thatincludes the necessary correlations for performing quantum errorcorrection, without the need to first generate a large entangled stateof qubits in an error correcting code. For example, embodimentsdisclosed herein described systems and methods whereby two-qubit (i.e.,joint) measurements, also referred to herein as “fusion measurements” or“fusion gates” can be performed on a collection of much smallerentangled states to generate a set of classical data that includes thelong-range correlations necessary to generate and decode the syndromegraph for a particular chosen cluster state, without the need toactually generate the cluster state. In other words, in some systems andmethods described herein, there is only ever generated a collection ofrelatively small entangled states (referred to herein as resourcestates) and then joint measurements are performed on these resourcestates directly to generate the syndrome graph data without the need tofirst generate (and then measure) a large cluster state that forms aquantum error correcting code (e.g., a topological code such as theRaussendorf lattice).

For example, as will be described in further detail below, in the caseof linear optical quantum computing using a Raussendorf lattice codestructure, to generate the syndrome graph data, a fusion gate can beapplied to a collection small entangled states (e.g., 4-GHZ states) thatare themselves not entangled with each other and thus are never part ofa larger Raussendorf lattice cluster state. Despite the fact that qubitsfrom the individual resource states were not mutually entangled prior tothe fusion measurement, the measurement outcomes that result from thefusion measurements generate a syndrome graph that includes all thenecessary correlations to perform quantum error correction. Such systemsand methods are referred to herein as Fusion Based Quantum Computing(FBQC). Advantageously, the resource states have a size that isindependent of the computation being performed or code distance used,which is in stark contrast to the cluster states of MBQC. This allowsthe resource states used for FBQC to be generated by a constant numberof sequential operations. As a result, in FBQC, errors in the resourcestate are bounded, which is important for fault-tolerance.

2. A SYSTEM FOR FBQC

FIG. 2 shows a quantum computing system in accordance with one or moreembodiments. The quantum computing system 201 includes a user interfacedevice 204 that is communicatively coupled to a quantum computing (QC)sub-system 206, described in more detail below in FIG. 3. The userinterface device 204 can be any type of user interface device, e.g., aterminal including a display, keyboard, mouse, touchscreen and the like.In addition, the user interface device can itself be a computer such asa personal computer (PC), laptop, tablet computer and the like. In someembodiments, the user interface device 204 provides an interface withwhich a user can interact with the QC subsystem 206 directly or via alocal area network, wide area network, or via the internet. For example,the user interface device 204 may run software, such as a text editor,an interactive development environment (IDE), command prompt, graphicaluser interface, and the like so that a user can program, or otherwiseinteract with, the QC subsystem to run one or more quantum algorithms.In other embodiments, the QC subsystem 206 may be pre-programmed and theuser interface device 204 may simply be an interface where a user caninitiate a quantum computation, monitor the progress, and receiveresults from the QC subsystem 206. QC subsystem 206 can further includea classical computing system 208 coupled to one or more quantumcomputing chips 210. In some examples, the classical computing system208 and the quantum computing chip 210 can be coupled to otherelectronic components 212, e.g., pulsed pump lasers, microwaveoscillators, power supplies, networking hardware, etc. In someembodiments that require cryogenic operation, the quantum computingsystem 201 can be housed within a cryostat, e.g., cryostat 214. In someembodiments, the quantum computing chip 210 can include one or moreconstituent chips, e.g., an integration (direct or heterogeneous) ofelectronic chip 216 and integrated photonics chip 218. Signals can berouted on- and off-chip any number of ways, e.g., via opticalinterconnects 220 and via other electronic interconnects 222. Inaddition, the computing system 201 may employ a quantum computingprocess, e.g., a fusion-based quantum computing process as described infurther detail below.

FIG. 3 shows a block diagram of a QC system 301 in accordance with someembodiments. Such a system can be associated with the computing system201 introduced above in reference to FIG. 2. In FIG. 3, solid linesrepresent quantum information channels and double-solid lines representclassical information channels. The QC system 301 includes a qubitentangling system 303, qubit fusion system 305, and classical computingsystem 307. In some embodiments, the qubit entangling system 303 cantake as input a collection of N physical qubits (also referred to hereinas “quantum sub-systems”), e.g., physical qubits 309 (also representedschematically as inputs 311 a, 311 b, 311 c, . . . , 311N) and cangenerate quantum entanglement between two or more of them to generateentangled resource states 315 (also referred to herein as “quantumsystems” which are themselves made up of entangled states of quantumsub-systems). For example, in the case of photonic qubits, the qubitentangling system 303 can be a linear optical system such as anintegrated photonic circuit that includes waveguides, beam splitters,photon detectors, delay lines, and the like. In some examples, theentangled resource states 315 can be relatively small entangled statesof qubits (e.g., qubit entangled states having between 3 and 30 qubits).In some embodiments, the resource states can be chosen such that thefusion operations applied to certain qubits of these states results insyndrome graph data that includes the required correlations for quantumerror correction. Advantageously, the system shown in FIG. 3 providesfor fault tolerant quantum computation using relatively small resourcestates, without requiring that the resource states become mutuallyentangled with each other to form the typical lattice cluster staterequired for MBQC.

In some embodiments, the input qubits 309 can be a collection of quantumsystems (also referred to herein as quantum-subsystems) and/or particlesand can be formed using any qubit architecture. For example, the quantumsystems can be particles such as atoms, ions, nuclei, and/or photons. Inother examples, the quantum systems can be other engineered quantumsystems such as flux qubits, phase qubits, or charge qubits (e.g.,formed from a superconducting Josephson junction), topological qubits(e.g., Majorana fermions), spin qubits formed from vacancy centers(e.g., nitrogen vacancies in diamond), or qubits otherwise encoded inmultiple quantum systems, e.g., Gottesman-Kitaev-Preskill (GKP) encodedqubits and the like. Furthermore, for the sake of clarity ofdescription, the term “qubit” is used herein although the system canalso employ quantum information carriers that encode information in amanner that is not necessarily associated with a binary bit. Forexample, qudits (i.e., quantum systems that can encode information inmore than two quantum states) can be used in accordance with someembodiments.

In accordance with some embodiments, the QC system 301 can be afusion-based quantum computer that can run one or more quantumalgorithms or software programs. For example, a software program (e.g.,a set of machine-readable instructions) that represents the quantumalgorithm to be run on the QC system 301 can be passed to a classicalcomputing system 307 (e.g., corresponding to system 208 in FIG. 2above). The classical computing system 307 can be any type of computingdevice such as a PC, one or more blade servers, and the like, or even ahigh-performance computing system such as a supercomputer, server farm,and the like. Such a system can include one or more processors (notshown) coupled to one or more computer memories, e.g., memory 306. Sucha computing system will be referred to herein as a “classical computer.”In some examples, the software program can be received by a classicalcomputing module, referred to herein as a fusion pattern generator 313.One function of the fusion pattern generator 313 is to generate a set ofmachine-level instructions from the input software program (which mayoriginate as high-level code that can be more easily written by a userto program the quantum computer).

In some embodiments, the fusion pattern generator 313 can operate as acompiler for software programs to be run on the quantum computer. Fusionpattern generator 313 can be implemented as pure hardware, puresoftware, or any combination of one or more hardware or softwarecomponents or modules. In various embodiments, fusion pattern generator313 can operate at runtime or in advance; in either case, machine-levelinstructions generated by fusion pattern generator 313 can be stored(e.g., in memory 306). In some examples, the compiled machine-levelinstructions take the form of one or more data frames that instruct thequbit fusion system 305 to make, at a given clock cycle of the quantumcomputer, one or more fusions between certain qubits from the separate,i.e., unentangled, resource states 315. For example, fusion pattern dataframe 317 is one example of the set of fusion measurements (e.g., TypeII fusion measurements, described in more detail below in reference toFIGS. 18-21) that should be applied between certain pairs of qubits fromdifferent entangled resource states 315 during a certain clock cycle asthe program is executed. In some embodiments, several fusion patterndata frames 317 can be stored in memory 306 as classical data. In someembodiments, the fusion pattern data frames 317 can dictate whether ornot XX Type II Fusion is to be applied (or whether any other type offusion, or not, is to be applied) for a particular fusion gate withinthe fusion array 321 of the qubit fusion system 305. In addition, thefusion pattern data frames 317 can indicate that the Type II fusion isto be performed in a different basis, e.g., XX, XY, ZZ, etc. As usedherein, the term XX Type II Fusion, YY Type II Fusion, XY Type IIFusion, ZZ Type II Fusion etc. refer to a fusion operation that appliesa particular a two-particle projective measurement, e.g., a Bellprojection which, depending on the Bell basis chosen, can project thetwo qubits onto one of the 4 Bell states. Such projective measurementsproduce two measurement outcomes that correspond to the eigenvalues ofthe corresponding pair of observables that are measured in the chosenbasis. For example, XX Fusion is a Bell projection that measures the XXand ZZ observables (each of which could have a +1 or −1 eigenvalue—or 0or 1 depending on the convention used), and XZ Fusions is a Bellprojection that measures the observable XZ and ZX observables, and thelike. FIGS. 18-21 below show example circuits for performing Type IIfusions for various choices of basis in a linear optical system butother Bell projective measurements are possible in other qubitarchitectures without departing from the scope of the presentdisclosure. One of ordinary skill will appreciate that in a linearoptical system, Type II Fusions perform probabilistic Bell measurements.FIGS. 18-21 discuss the probabilistic nature of linear optical fusion inthe context of fusion “success” and “failure” outcomes and will not berepeated here for the sake of clarity.

A fusion controller circuit 319 of the qubit fusion system 205 canreceive data that encodes the fusion pattern data frames 317 and, basedon this data, can generate configuration signals, e.g., analog and/ordigital electronic signals, that drive the hardware within the fusionarray 321. For example, for the case of photonic qubits, the fusiongates can include photon detectors coupled to one or more waveguides,beam splitters, interferometers, switches, polarizers, polarizationrotators and the like. More generally, the detectors can be any detectorthat can detect the quantum states of one or more of the qubits in theresource states 315. One of ordinary skill will appreciate that manytypes of detectors may be used depending on the particular qubitarchitecture being employed.

In some embodiments, the result of applying the fusion pattern dataframes 317 to the fusion array 321 is the generation of classical data(generated by the fusion gates' detectors) that is read out, andoptionally pre-processed, and sent to decoder 333. More specifically,the fusion array 321 can include a collection of measuring devices thatimplement the joint measurements between certain qubits from twodifferent resource states and generate a collection of measurementoutcomes associated with the joint measurement. These measurementoutcomes can be stored in a measurement outcome data frame, e.g., dataframe 322 and passed back to the classical computing system for furtherprocessing.

In some embodiments, any of the submodules in the QC system 301, e.g.,controller 323, quantum gate array 325, fusion array 321, fusioncontroller 319, fusion pattern generator 313, decoder 323, and logicalprocessor 308 can include any number of classical computing componentssuch as processors (CPUs, GPUs, TPUs) memory (any form of RAM, ROM),hard coded logic components (classical logic gates such as AND, OR, XOR,etc.) and/or programmable logic components such as field programmablegate arrays (FPGAs and the like). These modules can also include anynumber of application specific integrated circuits (ASICs),microcontrollers (MCUs), systems on a chip (SOCs), and other similarmicroelectronics.

In some embodiments, the entangled resource states 315 can be any typeof entangled resource state, that, when the fusion operations areperformed, produces measurement outcome data frames that include thenecessary correlations for performing fault tolerant quantumcomputation. While FIG. 3 shows an example of a collection of identicalresource states, a system can be employed that generates many differenttypes of resource states and can even dynamically change the type ofresource state being generated based on the demands of the quantumalgorithm being run. As described herein, the logical qubit measurementoutcomes 327 can be fault tolerantly recovered, e.g., via decoder 333,from the measurement outcomes 322 of the physical qubits. Logicalprocessor 308 can then process the logical outcomes as part of therunning of the program. As shown, the logical processor can feed backinformation to the fusion pattern generator 313 for affect downstreamgates and/or measurements to ensure that the computation proceeds faulttolerantly.

FIG. 4 illustrates an example of a qubit entangling system 401 inaccordance with some embodiments. Such a system can be used to generatequbits (e.g., photons) in an entangled state (e.g., the resource stateused in the illustrative examples shown in FIGS. 7-9 below), inaccordance with some embodiments. Qubit entangling system 401 is anexample of a system that can be employed in an FBQC system, such asqubit entangling system 303 shown in FIG. 3 above. One of ordinary skillwill appreciate that any qubit entangling system could be used withoutdeparting from the scope of the present disclosure. Examples of qubitentangling systems can be found in U.S. patent application Ser. No.16/621,994 (published as US Pat App Pub No 20200287631) titled,Generation of entangled qubit states, U.S. patent application Ser. No.16/691,459 (published as U.S. Pat. No. ______), titled, GENERATION OFENTANGLED PHOTONIC STATES, and U.S. patent application Ser. No.16/691,450 (published as U.S. Pat. No. ______), titled, GENERATION OF ANENTANGLED PHOTONIC STATE FROM PRIMITIVE RESOURCES, the disclosures ofwhich are hereby incorporated by reference in their entireties for allpurposes. For example, in some embodiments, rather than generatingsingle photons, the photon sources may generate entangled resourcestates directly, or may even generate smaller entangled states that canundergo additional entangling operations at the Entangled StateGenerator 400 to produce the final resource states to be used for FBQC.As such, as used herein the scope of the term “photon source” isintended to include at least sources of single photons, sources ofmultiple photons in entangled states, or more generally any source ofphotonic states. One of ordinary skill will appreciate that the preciseform of the resource state generation hardware is not critical and anysystem can be employed without departing from the scope of the presentdisclosure.

In an illustrative photonic architecture, qubit entangling system 401can include a photon source system 405 that is optically connected toentangled state generator 400. Both the photon source system 405 and theentangled state generator 400 may be coupled to a classical processingsystem 403 such that the classical processing system 403 can communicatewith and/or control (e.g., via the classical information channels 430a-b) the photon source system 405 and/or the entangled state generator400. Photon source system 405 may include a collection of single-photonsources that can provide output photonic states (e.g., single photons orother photonic states such as bel states, GHZ states, and the like) toentangled state generator 400 by way of interconnecting waveguides 402.Entangled state generator 400 may receive the output photonic states andconvert them to one or more entangled photonic states (or largerphotonic states in the case that the source itself outputs an entangledphotonic state) and then output these entangled photonic states intooutput waveguides 440. In some embodiments, output waveguides 440 can becoupled to some downstream circuit that may use the entangled states forperforming a quantum computation. For example, the entangled statesgenerated by the entangled state generator 400 may be used as resourcesfor a downstream quantum optical circuit (not shown).

In some embodiments, the photon source system 405 and the entangledstate generator 400 may be used in conjunction with the quantumcomputing system illustrated in FIG. 3. For example, the qubitentangling system 303 illustrated in FIG. 3 may include the photonsource system 405 and the entangled state generator 400, and theclassical computer system 403 of FIG. 4 may include one or more of thevarious classical computing components illustrated in FIG. 3 (e.g., theclassical computing system 307). In this case, the entangled photonsthat leave via output waveguides 440 can be fused together by the qubitfusion system 305, i.e., they can be input to a detection system thatperforms a collection of joint measurements for use in a FBQC scheme.

In some embodiments, system 401 may include classical channels 430(e.g., classical channels 430-a through 430-d) for interconnecting andproviding classical information between components. It should be notedthat classical channels 430-a through 430-d need not all be the same.For example, classical channel 430-a through 430-c may comprise abi-directional communication bus carrying one or more reference signals,e.g., one or more clock signals, one or more control signals, or anyother signal that carries classical information, e.g., heraldingsignals, photon detector readout signals, and the like.

In some embodiments, qubit entangling system 401 includes the classicalcomputer system 403 that communicates with and/or controls the photonsource system 405 and/or the entangled state generator 400. For example,in some embodiments, classical computer system 403 can be used toconfigure one or more circuits, e.g., using a system clock that may beprovided to photon sources 405 and entangled state generator 400 as wellas any downstream quantum photonic circuits used for performing quantumcomputation. In some embodiments, the quantum photonic circuits caninclude optical circuits, electrical circuits, or any other types ofcircuits. In some embodiments, classical computer system 403 includesmemory 404, one or more processor(s) 402, a power supply, aninput/output (I/O) subsystem, and a communication bus or interconnectingthese components. The processor(s) 402 may execute software modules,programs, and/or instructions stored in memory 404 and thereby performprocessing operations.

In some embodiments, memory 404 stores one or more programs (e.g., setsof instructions) and/or data structures. For example, in someembodiments, entangled state generator 400 can attempt to produce anentangled state over successive stages and/or over independentinstances, any one of which may be successful in producing an entangledstate. In some embodiments, memory 404 stores one or more programs fordetermining whether a respective stage was successful and configuringthe entangled state generator 400 accordingly (e.g., by configuringentangled state generator 400 to switch the photons to an output if thestage was successful, or pass the photons to the next stage of theentangled state generator 400 if the stage was not yet successful). Tothat end, in some embodiments, memory 404 stores detection patterns fromwhich the classical computing system 403 may determine whether a stagewas successful. In addition, memory 404 can store settings that areprovided to the various configurable components (e.g., switches)described herein that are configured by, e.g., setting one or more phaseshifts for the component.

In some embodiments, some or all of the above-described functions may beimplemented with hardware circuits on photon source system 405 and/orentangled state generator 400. For example, in some embodiments, photonsource system 405 includes one or more controllers 407-a (e.g., logiccontrollers) (e.g., which may comprise field programmable gate arrays(FPGAs), application specific integrated circuits (ASICS), a “system ona chip” that includes classical processors and memory, or the like). Insome embodiments, controller 407-a determines whether photon sourcesystem 405 was successful (e.g., for a given attempt on a given clockcycle) and outputs a reference signal indicating whether photon sourcesystem 405 was successful. For example, in some embodiments, controller407-a outputs a logical high value to classical channel 430-a and/orclassical channel 430-c when photon source system 405 is successful andoutputs a logical low value to classical channel 430-a and/or classicalchannel 430-c when photon source system 405 is not successful. In someembodiments, the output of control 407-a may be used to configurehardware in controller 107-b.

Similarly, in some embodiments, entangled state generator 400 includesone or more controllers 407-b (e.g., logical controllers) (e.g., whichmay comprise field programmable gate arrays (FPGAs), applicationspecific integrated circuits (ASICS), or the like) that determinewhether a respective stage of entangled state generator 400 hassucceeded, perform the switching logic described above, and output areference signal to classical channels 430-b and/or 430-d to informother components as to whether the entangled state generator 400 hassucceeded.

In some embodiments, a system clock signal can be provided to photonsource system 405 and entangled state generator 400 via an externalsource (not shown) or by classical computing system 403 via classicalchannels 430-a and/or 430-b. Examples of clock generators that may beused are described in U.S. Pat. No. 10,379,420, the contents of which ishereby incorporated by reference in its entirety for all purposes; butother clock generators may also be used without departing from the scopeof the present disclosure. In some embodiments, the system clock signalprovided to photon source system 405 triggers photon source system 405to attempt to output one photon per waveguide. In some embodiments, thesystem clock signal provided to entangled state generator 400 triggers,or gates, sets of detectors in entangled state generator 400 to attemptto detect photons. For example, in some embodiments, triggering a set ofdetectors in entangled state generator 400 to attempt to detect photonsincludes gating the set of detectors.

It should be noted that, in some embodiments, photon source system 405and entangled state generator 400 may have internal clocks. For example,photon source system 405 may have an internal clock generated and/orused by controller 407-a and entangled state generator 400 has aninternal clock generated and/or used by controller 407-b. In someembodiments, the internal clock of photon source system 405 and/orentangled state generator 400 is synchronized to an external clock(e.g., the system clock provided by classical computer system 403)(e.g., through a phase-locked loop). In some embodiments, any of theinternal clocks may themselves be used as the system clock, e.g., aninternal clock of the photon source may be distributed to othercomponents in the system and used as the master/system clock.

In some embodiments, photon source system 405 includes a plurality ofprobabilistic photon sources that may be spatially and/or temporallymultiplexed, i.e., a so-called multiplexed single photon source. In oneexample of such a source, the source is driven by a pump, e.g., a lightpulse, that is coupled into an optical resonator that, through somenonlinear process (e.g., spontaneous four wave mixing, second harmonicgeneration, and the like) may generate zero, one, or more photons. Asused herein, the term “attempt” is used to refer to the act of driving aphoton source with some sort of driving signal, e.g., a pump pulse, thatmay produce output photons non-deterministically (i.e., in response tothe driving signal, the probability that the photon source will generateone or more photons may be less than 1). In some embodiments, arespective photon source may be most likely to, on a respective attempt,produce zero photons (e.g., there may be a 90% probability of producingzero photons per attempt to produce a single-photon). The second mostlikely result for an attempt may be production of a single-photon (e.g.,there may be a 9% probability of producing a single-photon per attemptto produce a single-photon). The third most likely result for an attemptmay be production of two photons (e.g., there may be an approximately 1%probability of producing two photons per attempt to produce a singlephoton). In some circumstances, there may be less than a 1% probabilityof producing more than two photons.

In some embodiments, the apparent efficiency of the photon sources maybe increased by using a plurality of single-photon sources andmultiplexing the outputs of the plurality of photon sources. In someembodiments, the photon source can also produce a classical heraldsignal that announces (or heralds) the success of the generation. Insome embodiments, this classical signal is obtained from the output of adetector, where the photon source system always produces photon statesin pairs (such as in SPDC), and detection of one photon signal is usedto herald the success of the process. This herald signal can be providedto a multiplexer and used to properly route a successful generation to amultiplexer output port, as described in more detail below.

The precise type of photon source used is not critical and any type ofsource can be used, employing any photon generating process, such asspontaneous four wave mixing (SPFW), spontaneous parametricdown-conversion (SPDC), or any other process. Other classes of sourcesthat do not necessarily require a nonlinear material can also beemployed, such as those that employ atomic and/or artificial atomicsystems, e.g., quantum dot sources, color centers in crystals, and thelike. In some cases, sources may or may be coupled to photonic cavities,e.g., as can be the case for artificial atomic systems such as quantumdots coupled to cavities. Other types of photon sources also exist forSPWM and SPDC, such as optomechanical systems and the like. In someexamples the photon sources can emit multiple photons already in anentangled state in which case the entangled state generator 400 may notbe necessary, or alternatively may take the entangled states as inputand generate even larger entangled states.

In some embodiments, spatial multiplexing of several non-deterministicphoton sources (also referred to as a MUX photon source) can beemployed. Many different spatial MUX architectures are possible withoutdeparting from the scope of the present disclosure. Temporal MUXing canalso be implemented instead of or in combination with spatialmultiplexing. MUX schemes that employ log-tree, generalized Mach-Zehnderinterferometers, multimode interferometers, chained sources, chainedsources with dump-the-pump schemes, asymmetric multi-crystal singlephoton sources, or any other type of MUX architecture can be used. Insome embodiments, the photon source can employ a MUX scheme with quantumfeedback control and the like. One example of an n×m MUXed source isdisclosed in U.S. Pat. No. 10,677,985, the contents of which is herebyincorporated by reference in its entirety for all purposes.

FIG. 5 shows one example of qubit fusion system 501 in accordance withsome embodiments. In some embodiments, qubit fusion system 501 can beemployed within a larger FBQC system such as qubit fusion system 305shown in FIG. 3.

Qubit fusion system 501 includes a fusion controller 519 that is coupledto fusion array 521. Fusion controller 519 is configured to operate asdescribed above in reference to fusion controller circuit 319 of FIG. 3above. Fusion array 521 includes a collection of fusion sites that eachreceive two or more qubits from different resource states (not shown)and perform one or more fusion operations (e.g., Type II fusion) onselected qubits from the two or more resource states. The fusionoperations performed on the qubits can be controlled by the fusioncontroller 519 via classical signals that are sent from the fusioncontroller 519 to each of the fusion sites via control channels 503 a,503 b, etc. Based on the joint measurements performed at each fusionsite, classical measurement outcomes in the form of classical data areoutput and then provided to a decoder system, as shown and describedabove in reference to FIG. 3. Examples of photonic circuits that can beemployed as Type II fusion gates are described in below in reference toFIG. 6 and also FIGS. 18-20.

FIG. 6 shows one possible example of a fusion site 601 as configured tooperate with a fusion controller 319 to provide measurement outcomes toa decoder for fault tolerant quantum computation in accordance with someembodiments. In this example, fusion site 601 can be an element offusion array 321 (shown in FIG. 3), and although only one instance isshown for purposes of illustration, fusion array 321 can include anynumber of instances of fusion sites 601.

As described above, the qubit fusion system 305 can receive two or morequbits (Qubit 1 and Qubit 2, shown here in a dual rail encoding) thatare to be fused. Qubit 1 is one qubit that is entangled with one or moreother qubits (not shown) as part of a first resource state and Qubit 2is another qubit that is entangled with one or more other qubits (notshown) as part of a second resource state. Advantageously, in contrastto MBQC, none of the qubits from the first resource state need beentangled with any of the qubits from the second (or any other) resourcestate in order to facilitate a fault tolerant quantum computation. Alsoadvantageously, at the inputs of a fusion site 601, the collection ofresource states are not mutually entangled to form a cluster state thattakes the form of a quantum error correcting code and thus there is noneed to store and or maintain a large cluster state with long-rangeentanglement across the entire cluster state. Also advantageously, thefusion operations that take place at the fusion sites can be fullydestructive joint measurements on Qubit 1 and Qubit 2 such that all thatis left after the measurement is classical information representing themeasurement outcomes on the detectors, e.g., detectors 603, 605, 607,609. At this point, the classical information is all that is needed forthe decoder 333 to perform quantum error correction, and no furtherquantum information is propagated through the system. This can becontrasted with an MBQC system that might employ fusion sites to fuseresource states into a cluster state that itself serves as thetopological code and only then generates the required classicalinformation via single particle measurements on each qubit in the largecluster state. In such an MBQC system, not only does the large clusterstate need to be stored and maintained in the system before the singleparticle measurements are made, but an extra single particle measurementstep needs to be applied (in addition to the fusions used to generatethe cluster state) to every qubit of the cluster state in order togenerate the classical information required to compute the syndromegraph data required for the decoder to perform quantum error correction.

FIG. 6 shows an illustrative example for one way to implement a fusionsite as part of a photonic quantum computer architecture. In thisexample, qubit 1 and qubit 2 can be dual rail encoded photonic qubits. Abrief introduction to the dual-rail encoding of photonic qubits isprovided in Section 4 below, in reference to FIGS. 11-14. Accordingly,qubit 1 and qubit 2 can input on waveguides pair 621, 623 and onwaveguide pair 625, 627, respectively. Interferometers 624 and 628 canbe placed in line with each qubit, and within one arm of eachinterferometer 624, 628 a programmable phase shifter 630, 632 can beoptionally applied to affect the basis in which the fusion operation isapplied, e.g., by implementing the specific mode couplings shown in FIG.21 to implement what is referred to herein as XX, XY, YY, or ZZfusions). The programmable phase shifters 630, 632 can be coupled to thefusion controller 319 via control line 629 and 631 such that signalsfrom the fusion controller 319 can be used to set the basis in which thefusion operation is applied to the qubits. In some embodiments the basiscan be hard-coded within the fusion controller 319, or in someembodiments the basis can be chosen based upon external inputs, e.g.,instructions provided by the fusion pattern generator 313. Additionalmode couplers, e.g., mode couplers 633 and 632 can be applied after theinterferometers followed by single photon detectors 603, 605, 607, 609to provide a readout mechanism for performing the joint measurement.

In some embodiments, fusion can be probabilistic operation, i.e., itimplements a probabilistic Bell measurement, with the measurementsometimes succeeding and sometime failing, as described in FIG. 20below. In some embodiments, the success probability of such operationcan be increased by using extra quantum systems in addition to thoseonto which the operation is acting upon. Embodiments using extra quantumsystems are usually referred to as “boosted” fusion. In the exampleshown in FIG. 6, the fusion site implements an unboosted Type II fusionoperation on the incoming qubits. One of ordinary skill will appreciatethat any type of fusion operation can be applied (and may be boosted orunboosted) without departing from the scope of the present disclosure.Additional examples of Type II fusion circuits are shown and describedin Section 5 below for both polarization encoding and dual rail pathencoding. In some embodiments the fusion controller 319 can also providea control signal to the detector 603, 605, 607, 609. A control signalcan be used, e.g., for gating the detectors or for otherwise controllingthe operation of the detectors. Each of the detectors 603, 605, 607, 609provides photon detection signal (representing the number of photonsdetected by the detector, e.g., 0 photons detected, 1 photon detected,two photons detected, etc.), and this photon detection signal can bepreprocessed at the fusion site 601 to determine a measurement outcome(e.g., fusion success or not) or passed directly to the decoder 333 forfurther processing.

3. AN EXAMPLE OF FBQC EMPLOYING GHZ RESOURCE STATES

FIGS. 7A-7B illustrate an FBQC scheme for fault tolerant quantumcomputation in accordance with one or more embodiments. In this examplea topological code known as the Raussendorf lattice (also known as thefoliated surface code) is used but any other error correcting code canbe used without departing from the scope of the present disclosure. Forexample, FBQC can be implemented for various volume codes (such as thediamond code, triamond code, etc.), various color codes, or othertopological codes can be used without departing from the scope of thepresent disclosure.

FIG. 7A illustrates one unit cell 702 of a Raussendorf lattice. For thecase of measurement based quantum computing, to determine the value ofthe syndrome graph, referred to herein as P_(cell), at the center of theunit cell, the qubits on the six faces of the unit cell are measured inthe x-basis resulting in a set of 0 or 1 eigenvalues being determinedfor each of the six M_(x) measurements. These eigenvalues are thencombined as follows

$\begin{matrix}{{P_{{cell} - {MBQC}} = \lbrack {\sum\limits_{i = 1}^{6}{M_{x}( S_{i} )}} \rbrack}{mod}\; 2.} & (2)\end{matrix}$

where S₁, S₂, . . . , S₆ correspond to the six sites on the faces of theunit cell and M_(x)(S_(i)) corresponds to the measurement outcomes (0or 1) obtained by measuring the corresponding face qubits in the xbasis. (S₁, S₂, and S₃ are labeled in FIG. 7; S₄, S₅, and S₆ are locatedon the hidden faces of unit cell 702.)

In FBQC, the goal is to generate, through a series of joint measurements(e.g., a positive-operator valued measure, also referred to as a POVM)on two or more qubits, a set of classical data that corresponds to theerror syndrome of some quantum error correcting code. For example, usingthe Raussendorf unit cell of FIG. 7A as an illustrative example, the setof measurements that can be used to generate the syndrome graph value inan FBQC approach is shown in FIG. 7B. In this example, GHZ states areused as the resource states, but one of ordinary skill having thebenefit of this disclosure will appreciate that any suitable resourcestate can be used without departing from the scope of the presentdisclosure. To get from the MBQC scheme shown in FIG. 7A to the FBQCscheme shown in FIG. 7B, every face qubit of FIG. 7A is replaced withindividual qubits from distinctly separate (i.e., not entangled)resource states. For instance, four resource states R1, R2, and R3(encircled by dotted ellipses), are each contributing at least one qubitto what would be the face qubit S2 of the Raussendorf cell are labeledin FIG. 7B. For example, the face qubit S2 in FIG. 7A is replaced with 4qubits, from three different resource states: resource state R1contributes two qubits; resource state R2 contributes a third qubit; andresource state R3 contributes the fourth qubit. In operation, the systemwill perform two fusions at each face (e.g., circles 721, 722 in FIG. 7Brepresent fusions between the contributing qubits of resource state R2and R1 and R3 and R1, respectively). In an example where the fusions areType II fusions, all four face qubits are measured, thereby generatingfour measurement results. The syndrome graph value for the cell isobtained by Eq. (2) above, but now with

M _(x)(S _(i))=[F _(1,XX)(S _(i))+F _(2,XX)(S _(i))]mod 2  (3)

where, for the ith face, F_(1,XX)(S_(i)) is the measurement outcomeobtained by performing the joint measurement on the qubits associatedwith fusion 1 (e.g., as indicated by circle 721), with the fusion 1being a type II fusion performed in the XX basis and where F_(2,XX)(S_(i)) is the measurement outcome obtained by performing the jointmeasurement on the qubits associated with fusion 2 (e.g., as indicatedby circle 722), with the fusion 2 also being a type II fusion performedin the XX basis. Like the measurements associated with the X observabledescribed above in reference to Eqn. (2), the fusion measurements of theobservable XX (and ZZ) take the values of zero or 1 corresponding to thepositive or negative eigenvectors, respectively, of the measuredoperators (XX and ZZ, in this example). In view of Eq. (3), to obtaineach measurement on a face M_(x)(S_(i)), correct fusion outcomes forboth the fusion measurements F_(1,XX) (S_(i)) and F_(2,XX)(S_(i)) aredesired. However, if due to some error either fusions fails so thatvalues for the operators cannot be recovered, then in some embodiments,the measurement of the face is considered failed and results in at leastone erased edge in the syndrome graph data. One of ordinary skill havingthe benefit of this disclosure will appreciated that errors can be dealtwith by the decoder in a manner that is analogous to that describedabove in reference to FIGS. 1A-1C. One of ordinary skill in the art willalso recognize that while our description of Eqn. (3) focused on the XXobservable, the fusion can also produce the measurement of the ZZobservable and that those outcomes can also be combined as per Eqn. (3)to produce an independent set of syndrome graph date. In someembodiments these two sets of syndrome data are referred to as theprimal and dual syndrome graphs.

FIG. 7C shows an example of a cluster state made up on several unitcells of the Raussendorf lattice. In an MBQC approach, this entirecluster state would need to be generated, forming an entangled state ofmany qubits with the entanglement of the state extending across thelattice from one surface boundary to another. In the MBQC approach it isthis large entangled cluster state that serves as the quantum errorcorrecting code and thus can encode the logical qubit. Computationproceeds by performing single qubit measurements on each qubit of theentangled state to generate the measurement outcomes that are used togenerate the syndrome graph that is fed to the decoder as describedabove in reference to FIGS. 1A-1C. As such, increasing the errortolerance of the computation requires an increase to the size of thelattice and therefore an increase to the size of the entangled state. Inone or more embodiments of the FBQC approach disclose herein, such alarge entangled cluster state is not necessary, but rather, smallerresource states are generated, with the size of the resource statesbeing independent of the of the required error tolerance. As describedin detail above in reference to FIG. 7, the FBQC approach can beconstructed from any fault tolerant lattice by replacing each node ofthe lattice with a set of fusions between two or more adjacent resourcestates. This construction of replacing each node of the lattice with aresource state/fusions is merely one example of obtaining an FBQC schemeand one of ordinary skill having the benefit of this disclosure willrecognize that many different ways of constructing an FBQC scheme from afault tolerant lattice can be employed without departing from the scopeof the present disclosure.

Furthermore, as described in more detail below, the process can proceedby generating a layer of resource states in a given clock cycle andperforming fusions within each layer, as described in FIG. 8-9 below.For example, in FIG. 7C, the horizontal direction represents time in thesense that all or a subset of the qubits in any given layer in the x-yplane can be generated/initialized at the same clock cycle, e.g., qubitsin Layer 1 can be generated at clock cycle 1, qubits in layer 2 can begenerated at clock cycle 2, qubits in layer 3 can be generated at clockcycle 3, etc. As will be described in more detail below, a certainsubset qubits in each layer can be stored/delayed such that they areavailable to be fused with qubits from resource states in a subsequentlayer, if necessary to enable fault tolerance.

In some embodiments, in order to generate a desired error syndrome, alattice preparation protocol (LPP) can be designed that generates theappropriate syndrome graph from the fusions of multiple smallerentangled resource states. FIGS. 8-9 show an example of a latticepreparation protocol according to some embodiments. For the purposes ofillustration, the resource states are states such as resource state 800shown in FIG. 8A; however other resource states can be used withoutdeparting from the scope of the present disclosure. The resource state800 is equivalent to a GHZ state up to the application of Hadamard gatesto single qubits. For example, the states used in the example disclosedherein are equivalent to GHZ states up to the application of Hadamardgates to the two terminating end qubits 800 a-3 and 800 a-4 in FIG. 8A.More specifically, a 4-GHZ state can be identified as a stabilizer statehaving the following stabilizers:

XXXX, ZZII,ZIZI,ZIIZ

. The resource state 800 shown in FIG. 8A is closely related to this GHZstate, but the stabilizers of state 800 are

XXZZ, ZZII, ZIXI, ZIIX

(with the ordering of the operators corresponding to qubits 800 a-1, 800a-2, 800 a-3, and 800 a-4, respectively). One of ordinary skill willappreciate that 4-GHZ state and the resource state 800 are equivalentunder the application of a Hadamard gate on qubits 800-a3 and 800-a4.

The time direction in FIGS. 8-9 is perpendicular to the page such that aresource state having a shape such as resource state 810 represents acollection of qubits, qubits 1, 2, and 3 that are mutually entangledwithin the same clock cycle and qubit 4 which is entangled in the timedimension with, e.g., qubits 2 and 3. Such a resource state can becreated by, e.g., generating the full 4 qubit resource state in a singleclock cycle and then storing qubit 4 for a fixed time period (e.g., oneclock cycle) in a memory. As used herein the term “memory” includes atany type of memory, e.g., a quantum memory, a qubit delay line, a shiftregister for qubits, a qubit itself, and the like. In the case ofphotonic resource states, qubit memories such as these are equivalent toqubit delays and can thus be implemented through the use of opticalfiber. In the example shown in FIG. 8C, the delay to qubit 4 isrepresented schematically by a loop of additional optical path length(e.g. provided by an optical fiber) placed inline with the existingoptical path of the qubit but that is not present in the optical path ofqubits 1-3. In this example the length of the fiber is such that itimplements a single clock cycle delay of duration T but other delays arepossible as well, e.g., 2T, 3T, etc. In terms of physical delay times,such delays could be in the range of 500 ps-500 ns but any delay ispossible without departing from the scope of the present disclosure.

Returning to the FBQC process disclosed herein, FIGS. 8-9 show anexample of how a lattice preparation and measurement protocol for FBQCcan proceed according to layers. FIG. 8A shows a portion of theunderlying layer of the Raussendorf lattice shown as layer 810(corresponding to a portion of Layer 1 shown in FIG. 7C). In the exampleillustrated here, to process a layer like that shown in FIG. 8A, firstmultiple resource states 800 are generated (e.g., in qubit entanglingsystem 303 of FIG. 3). In this example, the resource state 800 is anentangled state comprising 4 physical qubits (also referred to herein asquantum sub-systems): qubits 800 a-1, 800 a-2, 800 a-3, 800 a-4. In someembodiments, the resource state 800 can take the form of a 4-GHZ statewhere the two terminating end qubits 800 a-4 and 800 a-3 have undergonea Hadamard operation (e.g., for the case of a dual rail encoded qubit,by way of applying a 50:50 beamsplitter between the two rails that formthe qubit). In some embodiments, not all qubits in the layer are subjectto fusions in this clock cycle, but rather some of the qubits generatedduring this clock cycle from certain resource states can be delayed,e.g., the measurement of qubit 820, redundantly encoded qubit 805, orany other qubit can be delayed so that the qubit will be available atthe next clock cycle. Such delayed qubits are then available to be fusedwith one or more qubits from resource states that will only be availablefor fusions at the next clock cycle.

In examples that employ a photonic implementation, the qubits from theresource states can then be routed appropriately (via integratedwaveguides, optical fiber, or any other suitable photonic routingtechnology) to the qubit fusion system (e.g., qubit fusion system 305 ofFIG. 3) to enable a set of fusion measurements that implement quantumerror correction, i.e., that will result in collecting the measurementoutcomes that correspond to the error syndrome of choice. While thisexample explicitly uses a topological code based on the Raussendorflattice, any code can be used without departing from the scope of thepresent disclosure.

FIG. 8B shows an example of a collection of GHZ resource statesarranged, i.e., that they have been pre-routed, such that the qubitsthat are to be sent to a given fusion gate are positioned graphicallyadjacent to each other. For qubits that are adjacent to each other inthis illustration, respective fusions can be performed between pairs ofqubits (also referred to herein as respective quantum sub-systems, witheach qubit from the pair of qubits input on a fusion site belonging to adifferent respective resource state). For example, at site 802, two TypeII fusion measurements can be applied, one between qubits 822 and 824and one between qubits 826 and 828. It should be noted that before thefusions are performed qubits 822 and 824 (or qubits 826 and 828) are notentangled with one another but instead are each part of a distinctresource state. As such, the large entangled cluster state known as theRaussendorf lattice is not present before the fusion measurements areperformed.

Referring to FIG. 9A, a portion of a second layer of the underlying codestructure is shown as layer 910 (corresponding to layer 2 shown in FIG.7C). In an FBQC system, to process a single layer like that shown inFIG. 9B the FBQC method proceeds along the same lines as described abovein reference to FIGS. 8A-8B so the details will not be repeated here.

FIGS. 10A-10E show in further detail a method for performing FBQC inaccordance with one or more embodiments. More specifically, the methoddescribed here includes steps for performing the joint measurements fora particular quantum error correcting code according to someembodiments, where the different layers of the code may be generated atdifferent time steps (clock cycles) as introduced above in reference toFIGS. 8-9 and entangled together in a manner that provides for fusionsmeasurements to extract the necessary syndrome information forperforming quantum error correction. Like other examples providedherein, the Raussendorf lattice is used for the sake of illustration butother codes can be used without departing from the scope of the presentdisclosure.

For example, FIGS. 10A and 10B shows portions of layers 1 and 3, andlayers 2 and 4, respectively, from the Raussendorf lattice of FIG. 7C(referred to here as the quantum error correcting (QEC) code). FIGS. 10Cand 10D illustrate a method for processing these layers in an FBQCsystem, including example resource states that could be used. For thesake of example, the description is limited to vertices 1, 2, 3, and 4of the QEC code and the example focuses on how to perform the resourcestate generation and measurements in an FBQC system.

Returning to FIG. 10A, in step 1001, a first set of the resource statesare provided during a first clock cycle. FIG. 10D shows one examplewhere, instead of single qubits being provided at vertices 1, 2, 3, 4,5, etc. with those single qubits being mutually entangled across thelattice (as would be the case for a MBQC system), two or more qubits areprovided, each originating from different, non-entangled, resourcestates, e.g., respective resource states A, B, C, D, E, F, and G. Asused herein, the notation A^(i) _(j) is used to denote the j-th qubitfrom the A-th resource state of the i-th layer. For example, the A-thresource state of layer 1 in FIG. 10D is a GHZ state that includes 4qubits, labeled A¹ ₁, A¹ ₂, A¹ ₃, A¹ ₄ as shown. Likewise, the qubitscomprising resource state B that is provided as part of layer 1 can belabeled as B¹ ₁, B¹ ₂, B¹ ₃, B¹ ₄ (but this time with labels notexplicitly shown in the figure to avoid cluttering the diagram). Qubitsthat will be fused to generate the syndrome information associated withvertices 1, 2, 3, 4, 5 are also shown as enclosed by solid ellipses 1,2, 3, and 4 in FIG. 10D. As used herein these vertices are eachassociated with hardware for performing Type II fusions at fusion sites,as described above in reference to FIGS. 3-6.

In some embodiments, the resource states for any given layer can begenerated/provided by a qubit entangling system such as that describedabove in reference to FIGS. 3 and 4. However, one of ordinary skillhaving the benefit of this disclosure will understand that any qubitentangling system can be employed, and that a given qubit entanglingsystem can employ many different types of resource state generators,even generating different types of resource states. In this sense, anFBQC system is completely agnostic to the choice of resource states andchoice of architecture for the qubit entangling system, or even thearchitecture of the qubit itself, thereby leaving the system designer agreat deal of flexibility to implement a system that results in thehighest threshold for the given the prevailing error/noise sources.

In Step 1003, fusion instructions in the form of classical data (alsoreferred to herein as a fusion pattern) are provided to the fusionsites. Referring back to FIG. 3, for example, fusion pattern data frame317 is one example of the set of fusion instructions (e.g., Type IIfusion measurements in the XX basis) that can be applied between pairsof qubits from different entangled resource states at a fusion siteduring a certain clock cycle as a quantum application is executed on theFBQC system. As also described above, in some embodiments, severalfusion pattern data frames can be stored in memory as classical data. Insome embodiments, the fusion pattern data frames can dictate whether ornot XX Type II Fusion is to be applied (or whether any other type offusion, or not, is to be applied) for a particular fusion gate withinthe fusion site. In addition, the fusion pattern data frames canindicate that the Type II fusion is to be performed in a differentbasis, e.g., XX, XY, ZZ, etc.

Returning to FIG. 10D, the fusion instructions for Layer 1 can includefusion parameters (qubit location and basis) to fuse two or more qubitsfrom different resource states (also referred to herein as respectivequantum sub-systems because the qubits reside in, or are part of,respectively separate resource states). For example, for fusion site 1the fusion instructions can specify the fusion parameters to indicatethat XX Type II Fusions are to be performed between qubits from resourcestates A¹, B¹, and C¹ (and similarly for site 3 between E¹, F¹, and G¹).More specifically, the two Type II Fusions to be performed at fusionsite 1 can be specified to be between A¹ ₄ and B¹ ₂ and between C¹ ₁ andB¹ ₃. Similar instructions are provided for the other fusion sites inthe layer. For example, for fusion site 2, the fusion instructions canspecify the fusion parameters to indicate that XX Type II Fusions are tobe performed between qubits from resource states B¹, D¹, and F¹. Morespecifically, the two Type II Fusions to be performed at fusion site 2can be specified to be between B¹ ₄ and D¹ ₂ and between D¹ ₃ and F¹ ₄.However, unlike the case for fusion site 1, where all the qubits weremeasured, fusion site 2 includes a qubit that is to remain unmeasureduntil the second clock cycle. This is because the underlying structureof the QEC lattice requires that the quantum state of this qubit to bepreserved until it is to be fused to a qubit from a different layer at adifferent clock cycle, i.e., if this were an MBQC scheme the qubitassociated with this vertex would be one that is entangled with qubit inanother layer, e.g., qubits 2 and 6 shown in FIGS. 10B and 10C,respectively.

Returning to the explicit example shown in FIG. 10D, the fusioninstructions can specify that D¹ ₄ will not be measured until the nextclock cycle, where it will be fused from qubits in a later layer, e.g.,layer 2 shown in FIG. 10E. In a photonic implementation optical fibercan implement a qubit delay for the above function, serving as areliable quantum memory to store qubits until they are needed for afuture clock cycle. As used herein, these unmeasured (delayed) qubitsare referred to as unmeasured quantum sub-systems.

Moving on to fusion site 4, this site is an example that includesfusions between layers, i.e., fusion between qubits from resource statesthat were generated in this clock cycle with qubits from resource statesthat were generated in a prior clock cycle but were not measured at thattime but instead were delayed, or equivalently, stored until the nextclock cycle. For fusion site 4, the fusion instructions can specify thefusion parameters to indicate that XX Type II Fusions are to beperformed between qubits from resource states in three different layersC¹, B⁰, and B². The fusion instructions can also include instructions todelay (not measure) qubits C¹ ₂ and C¹ ₃ until the next clock cycle. Forexample, in this case, the fusion instructions can indicate that in thenext time step, C¹ ₂ is to be fused with B⁰ ₄ and C¹ ₃ is to be fusedwith B² ₁.

In Step 1003, the fusion operations that are specified by the fusioninstructions are performed, thereby generating classical data in theform of fusion measurement outcomes. As described above in reference toFIGS. 3-6 and Eq. (2), this classical data is then passed to the decoderand is used to construct the syndrome graph to be used for quantum errorcorrection.

These examples are illustrative. The choice of error correcting codedetermines the set of qubit pairs that are fused from certain resourcestates, such that the output of the qubit fusion system is the classicaldata from which the syndrome graph can be directly constructed. In someembodiments, the classical error syndrome data is generated directlyfrom the qubit fusion system without the need to preform additionalsingle particle measurements on any remaining qubits. In someembodiments, the joint measurements performed at the qubit fusion systemare destructive of the qubits upon which joint measurement is performed.

4. INTRODUCTION TO QUBITS AND PATH ENCODING

The dynamics of quantum objects, e.g., photons, electrons, atoms, ions,molecules, nanostructures, and the like, follow the rules of quantumtheory. More specifically, in quantum theory, the quantum state of aquantum object, e.g., a photon, is described by a set of physicalproperties, the complete set of which is referred to as a mode. In someembodiments, a mode is defined by specifying the value (or distributionof values) of one or more properties of the quantum object. For example,again for photons, modes can be defined by the frequency of the photon,the position in space of the photon (e.g., which waveguide orsuperposition of waveguides the photon is propagating within), theassociated direction of propagation (e.g., the k-vector for a photon infree space), the polarization state of the photon (e.g., the direction(horizontal or vertical) of the photon's electric and/or magneticfields) and the like.

For the case of photons propagating in a waveguide, it is convenient toexpress the state of the photon as one of a set of discretespatio-temporal modes. For example, the spatial mode k_(i) of the photonis determined according to which one of a finite set of discretewaveguides the photon can be propagating in. Furthermore, the temporalmode t_(j) is determined by which one of a set of discrete time periods(referred to herein as “bins”) the photon can be present in. In someembodiments, the temporal discretization of the system can be providedby the timing of a pulsed laser which is responsible for generating thephotons. In the examples below, spatial modes will be used primarily toavoid complication of the description. However, one of ordinary skillwill appreciate that the systems and methods can apply to any type ofmode, e.g., temporal modes, polarization modes, and any other mode orset of modes that serves to specify the quantum state. Furthermore, inthe description that follows, embodiments will be described that employphotonic waveguides to define the spatial modes of the photon. However,one of ordinary skill having the benefit of this disclosure willappreciate that any type of mode, e.g., polarization modes, temporalmodes, and the like, can be used without departing from the scope of thepresent disclosure.

For quantum systems of multiple indistinguishable particles, rather thandescribing the quantum state of each particle in the system, it isuseful to describe the quantum state of the entire many-body systemusing the formalism of Fock states (sometimes referred to as theoccupation number representation). In the Fock state description, themany-body quantum state is specified by how many particles there are ineach mode of the system. Because modes are the complete set ofproperties, this description is sufficient. For example, a multi-mode,two particle Fock state |1001

_(1,2,3,4) specifies a two-particle quantum state with one photon inmode 1, zero photons in mode 2, zero photons in mode three, and 1 photonin mode four. Again, as introduced above, a mode can be any set ofproperties of the quantum object (and can depend on the single particlebasis states being used to define the quantum state). For the case ofthe photon, any two modes of the electromagnetic field can be used,e.g., one may design the system to use modes that are related to adegree of freedom that can be manipulated passively with linear optics.For example, polarization, spatial degree of freedom, or angularmomentum, could be used. For example, the four-mode system representedby the two particle Fock state |1001

_(1,2,3,4) can be physically implemented as four distinct waveguideswith two of the four waveguides (representing mode 1 and mode 4,respectively) having one photon travelling within them. Other examplesof a state of such a many-body quantum system are the four photon Fockstate |1111

_(1,2,3,4) that represents each waveguide containing one photon and thefour photon Fock state |2200

_(1,2,3,4) that represents waveguides one and two respectively housingtwo photons and waveguides three and four housing zero photons. Formodes having zero photons present, the term “vacuum mode” is used. Forexample, for the four photon Fock state |2200

_(1,2,3,4) modes 3 and 4 are referred to herein as “vacuum modes” (alsoreferred to as “ancilla modes”).

As used herein, a “qubit” (or quantum bit) is a physical quantum systemwith an associated quantum state that can be used to encode information.Qubits, in contrast to classical bits, can have a state that is asuperposition of logical values such as 0 and 1. In some embodiments, aqubit is “dual-rail encoded” such that the logical value of the qubit isencoded by occupation of one of two modes by exactly one photon (asingle photon). For example, consider the two spatial modes of aphotonic system associated with two distinct waveguides. In someembodiments, the logical 0 and 1 values can be encoded as follows:

|0

_(L)=|10

_(1,2)  (1)

|1

_(L)=|01

_(1,2)  (2)

where the subscript “L” indicates that the ket represents a logicalvalue (e.g., a qubit value) and, as before, the notation |ij

_(1,2) on the right-hand side of the Equations (1)-(2) above indicatesthat there are i photons in a first waveguide and j photons in a secondwaveguide, respectively (e.g., where i and j are integers). In thisnotation, a two qubit state having a logical value |01

_(L) (representing a state of two qubits, the first qubit being in a ‘0’logical state and the second qubit being in a ‘1’ logical state) may berepresented using photon occupations across four distinct waveguides by|1001

_(1,2,3,4) (i.e., one photon in a first waveguide, zero photons in asecond waveguide, zero photons in a third waveguide, and one photon in afourth waveguide). In some instances, throughout this disclosure, thevarious subscripts are omitted to avoid unnecessary mathematicalclutter.

5. LOQC INTRODUCTION

5.1. Dual Rail Photonic Qubits

Qubits (and operations on qubits) can be implemented using a variety ofphysical systems. In some examples described herein, qubits are providedin an integrated photonic system employing waveguides, beam splitters(or directional couplers), photonic switches, and single photondetectors, and the modes that can be occupied by photons arespatiotemporal modes that correspond to presence of a photon in awaveguide. Modes can be coupled using mode couplers, e.g., optical beamsplitters, to implement transformation operations, and measurementoperations can be implemented by coupling single-photon detectors tospecific waveguides. One of ordinary skill in the art with access tothis disclosure will appreciate that modes defined by any appropriateset of degrees of freedom, e.g., polarization modes, temporal modes, andthe like, can be used without departing from the scope of the presentdisclosure. For instance, for modes that only differ in polarization(e.g., horizontal (H) and vertical (V)), a mode coupler can be anyoptical element that coherently rotates polarization, e.g., abirefringent material such as a waveplate. For other systems such as iontrap systems or neutral atom systems, a mode coupler can be any physicalmechanism that can couple two modes, e.g., a pulsed electromagneticfield that is tuned to couple two internal states of the atom/ion.

In some embodiments of a photonic quantum computing system usingdual-rail encoding, a qubit can be implemented using a pair ofwaveguides. FIG. 11A shows two representations (1100, 1100′) of aportion of a pair of waveguides 1102, 1104 that can be used to provide adual-rail-encoded photonic qubit. At 1100, a photon 1106 is in waveguide1102 and no photon is in waveguide 1104 (also referred to as a vacuummode); in some embodiments, this corresponds to the |0

state of a photonic qubit. At 1100′, a photon 1108 is in waveguide 1104,and no photon is in waveguide 1102; in some embodiments this correspondsto the |1

state of the photonic qubit. To prepare a photonic qubit in a knownstate, a photon source (not shown) can be coupled to one end of one ofthe waveguides. The photon source can be operated to emit a singlephoton into the waveguide to which it is coupled, thereby preparing aphotonic qubit in a known state. Photons travel through the waveguides,and by periodically operating the photon source, a quantum system havingqubits whose logical states map to different temporal modes of thephotonic system can be created in the same pair of waveguides. Inaddition, by providing multiple pairs of waveguides, a quantum systemhaving qubits whose logical states correspond to differentspatiotemporal modes can be created. It should be understood that thewaveguides in such a system need not have any particular spatialrelationship to each other. For instance, they can be but need not bearranged in parallel.

Occupied modes can be created by using a photon source to generate aphoton that then propagates in the desired waveguide. A photon sourcecan be, for instance, a resonator-based source that emits photon pairs,also referred to as a heralded single photon source. In one example ofsuch a source, the source is driven by a pump, e.g., a light pulse, thatis coupled into a system of optical resonators that, through a nonlinearoptical process (e.g., spontaneous four wave mixing (SFWM), spontaneousparametric down-conversion (SPDC), second harmonic generation, or thelike), can generate a pair of photons. Many different types of photonsources can be employed. Examples of photon pair sources can include amicroring-based spontaneous four wave mixing (SPFW) heralded photonsource (HPS). However, the precise type of photon source used is notcritical and any type of source, employing any process, such as SPFW,SPDC, or any other process can be used. Other classes of sources that donot necessarily require a nonlinear material can also be employed, suchas those that employ atomic and/or artificial atomic systems, e.g.,quantum dot sources, color centers in crystals, and the like. In somecases, sources may or may not be coupled to photonic cavities, e.g., ascan be the case for artificial atomic systems such as quantum dotscoupled to cavities. Other types of photon sources also exist for SPWMand SPDC, such as optomechanical systems and the like.

In such cases, operation of the photon source may be deterministic ornon-deterministic (also sometimes referred to as “stochastic”) such thata given pump pulse may or may not produce a photon pair. In someembodiments, coherent spatial and/or temporal multiplexing of severalnon-deterministic sources (referred to herein as “active” multiplexing)can be used to allow the probability of having one mode become occupiedduring a given cycle to approach 1. One of ordinary skill willappreciate that many different active multiplexing architectures thatincorporate spatial and/or temporal multiplexing are possible. Forinstance, active multiplexing schemes that employ log-tree, generalizedMach-Zehnder interferometers, multimode interferometers, chainedsources, chained sources with dump-the-pump schemes, asymmetricmulti-crystal single photon sources, or any other type of activemultiplexing architecture can be used. In some embodiments, the photonsource can employ an active multiplexing scheme with quantum feedbackcontrol and the like.

Measurement operations can be implemented by coupling a waveguide to asingle-photon detector that generates a classical signal (e.g., adigital logic signal) indicating that a photon has been detected by thedetector. Any type of photodetector that has sensitivity to singlephotons can be used. In some embodiments, detection of a photon (e.g.,at the output end of a waveguide) indicates an occupied mode whileabsence of a detected photon can indicate an unoccupied mode. In someembodiments, a measurement operation is performed in a particular basis(e.g., a basis defined by one of the Pauli matrices and referred to asX, Y, or Z), and mode coupling as described below can be applied totransform a qubit to a particular basis.

Some embodiments described below relate to physical implementations ofunitary transform operations that couple modes of a quantum system,which can be understood as transforming the quantum state of the system.For instance, if the initial state of the quantum system (prior to modecoupling) is one in which one mode is occupied with probability 1 andanother mode is unoccupied with probability 1 (e.g., a state |10

in a Fock notation in which the numbers indicate occupancy of eachstate), mode coupling can result in a state in which both modes have anonzero probability of being occupied, e.g., a state a₁|10

+a₂|01

, where |a₁|²+|a₂|²=1. In some embodiments, operations of this kind canbe implemented by using beam splitters to couple modes together andvariable phase shifters to apply phase shifts to one or more modes. Theamplitudes a₁ and a₂ depend on the reflectivity (or transmissivity) ofthe beam splitters and on any phase shifts that are introduced.

FIG. 11B shows a schematic diagram 1110 (also referred to as a circuitdiagram or circuit notation) for coupling of two modes. The modes aredrawn as horizontal lines 1112,1114, and the mode coupler 1116 isindicated by a vertical line that is terminated with nodes (solid dots)to identify the modes being coupled. In the more specific language oflinear quantum optics, the mode coupler 1116 shown in FIG. 11Brepresents a 50/50 beam splitter that implements a transfer matrix:

$\begin{matrix}{{T = {\frac{1}{\sqrt{2}}\begin{pmatrix}1 & i \\i & 1\end{pmatrix}}},} & (4)\end{matrix}$

where T defines the linear map for the photon creation operators on twomodes. (In certain contexts, transfer matrix T can be understood asimplementing a first-order imaginary Hadamard transform.) By conventionthe first column of the transfer matrix corresponds to creationoperators on the top mode (referred to herein as mode 1, labeled ashorizontal line 1112), and the second column corresponds to creationoperators on the second mode (referred to herein as mode 2, labeled ashorizontal line 1114), and so on if the system includes more than twomodes. More explicitly, the mapping can be written as:

$\begin{matrix}{ \begin{pmatrix}a_{1}^{\dagger} \\a_{2}^{\dagger}\end{pmatrix}_{input}\mapsto{\frac{1}{\sqrt{2}}\begin{pmatrix}1 & {- i} \\{- i} & 1\end{pmatrix}\begin{pmatrix}a_{1}^{\dagger} \\a_{2}^{\dagger}\end{pmatrix}_{output}} ,} & (5)\end{matrix}$

where subscripts on the creation operators indicate the mode that isoperated on, the subscripts input and output identify the form of thecreation operators before and after the beam splitter, respectively andwhere:

a _(i) |n _(i) ,n _(j)

=√{square root over (n _(i))}|n _(i)−1,n _(j)

a _(j) |n _(i) ,n _(j)

=√{square root over (n _(j))}|n _(i) ,n _(j)−1

a _(j) ^(†) |n _(i) ,n _(j)

=√{square root over (n _(j)+1)}|n _(i) ,n _(j)+1

  (6)

For example, the application of the mode coupler shown in FIG. 11B leadsto the following mappings:

$\begin{matrix}{ a_{1_{input}}^{\dagger}\mapsto{\frac{1}{\sqrt{2}}( {a_{1_{output}}^{\dagger} - {ia_{2_{output}}^{\dagger}}} )}  a_{2_{input}}^{\dagger}\mapsto{\frac{1}{\sqrt{2}}( {{{- i}a_{1_{output}}^{\dagger}} + a_{2_{output}}^{\dagger}} )} } & (7)\end{matrix}$

Thus, the action of the mode coupler described by Eq. (4) is to take theinput states |10

,|01

, and |11

to

$\begin{matrix}      {  | {10}  \rangle\mapsto\frac{ { | {10}  \rangle - i} \middle| {01} \rangle}{\sqrt{2}} ❘01} \rangle\mapsto\frac{ { {- i} \middle| {10} \rangle +} \middle| {01} \rangle}{\sqrt{2}}  \middle| 11 \rangle\mapsto{{\frac{- i}{2}( | {20}  \rangle} +}  \middle| 02 \rangle ) & (8)\end{matrix}$

FIG. 11C shows a physical implementation of a mode coupling thatimplements the transfer matrix T of Eq. (4) for two photonic modes inaccordance with some embodiments. In this example, the mode coupling isimplemented using a waveguide beam splitter 1120, also sometimesreferred to as a directional coupler or mode coupler. Waveguide beamsplitter 1120 can be realized by bringing two waveguides 1122, 1124 intoclose enough proximity that the evanescent field of one waveguide cancouple into the other. By adjusting the separation d between waveguides1122, 1124 and/or the length l of the coupling region, differentcouplings between modes can be obtained. In this manner, a waveguidebeam splitter 1120 can be configured to have a desired transmissivity.For example, the beam splitter can be engineered to have atransmissivity equal to 0.5 (i.e., a 50/50 beam splitter forimplementing the specific form of the transfer matrix T introducedabove). If other transfer matrices are desired, the reflectivity (or thetransmissivity) can be engineered to be greater than 0.6, greater than0.7, greater than 0.8, or greater than 0.9 without departing from thescope of the present disclosure.

In addition to mode coupling, some unitary transforms may involve phaseshifts applied to one or more modes. In some photonic implementations,variable phase-shifters can be implemented in integrated circuits,providing control over the relative phases of the state of a photonspread over multiple modes. Examples of transfer matrices that definesuch a phase shifts are given by (for applying a +i and −i phase shiftto the second mode, respectively):

$\begin{matrix}{{s = \begin{pmatrix}1 & 0 \\0 & i\end{pmatrix}}{s^{\dagger} = \begin{pmatrix}1 & 0 \\0 & {- i}\end{pmatrix}}} & (9)\end{matrix}$

For silica-on-silicon materials some embodiments implement variablephase-shifters using thermo-optical switches. The thermo-opticalswitches use resistive elements fabricated on the surface of the chip,that via the thermo-optical effect can provide a change of therefractive index n by raising the temperature of the waveguide by anamount of the order of 10-5 K. One of skill in the art with access tothe present disclosure will understand that any effect that changes therefractive index of a portion of the waveguide can be used to generate avariable, electrically tunable, phase shift. For example, someembodiments use beam splitters based on any material that supports anelectro-optic effect, so-called x2 and x3 materials such as lithiumniobite, BBO, KTP, BTO, PZT, and the like and even doped semiconductorssuch as silicon, germanium, and the like.

5.2. Photonic Mode Coupler: Beam Splitters

Beam splitters with variable transmissivity and arbitrary phaserelationships between output modes can also be achieved by combiningdirectional couplers and variable phase-shifters in a Mach-ZehnderInterferometer (MZI) configuration 1130, e.g., as shown in FIG. 11D.Complete control over the relative phase and amplitude of the two modes1132 a, 1132 b in dual rail encoding can be achieved by varying thephases imparted by phase shifters 1136 a, 1136 b, and 1136 c and thelength and proximity of coupling regions 1134 a and 1134 b. FIG. 11Eshows a slightly simpler example of a MZI 1140 that allows for avariable transmissivity between modes 1132 a, 1132 b by varying thephase imparted by the phase shifter 1137. FIGS. 11D and 11E are examplesof how one could implement a mode coupler in a physical device, but anytype of mode coupler/beam splitter can be used without departing fromthe scope of the present disclosure.

In some embodiments, beam splitters and phase shifters can be employedin combination to implement a variety of transfer matrices. For example,FIG. 12A shows, in a schematic form similar to that of FIG. 11A, a modecoupler 1200 implementing the following transfer matrix:

$\begin{matrix}{{T_{r} = {\frac{1}{\sqrt{2}}\begin{pmatrix}1 & 1 \\1 & {- 1}\end{pmatrix}}}.} & (10)\end{matrix}$

Thus, mode coupler 1200 applies the following mappings:

$\begin{matrix}{      {  | {10}  \rangle\mapsto\frac{ { | {10}  \rangle +} \middle| {01} \rangle}{\sqrt{2}} ❘01} \rangle\mapsto\frac{ { | {10}  \rangle -} \middle| {01} \rangle}{\sqrt{2}}  \middle| 11 \rangle\mapsto{{\frac{1}{2}( | {20}  \rangle} +}  \middle| 02 \rangle ).} & (11)\end{matrix}$

The transfer matrix T_(r) of Eq. (10) is related to the transfer matrixT of Eq. (4) by a phase shift on the second mode. This is schematicallyillustrated in FIG. 12A by the closed node 1207 where mode coupler 1216couples to the first mode (line 1212) and open node 1208 where modecoupler 1216 couples to the second mode (line 1214). More specifically,T_(r)=sTs, and, as shown at the right-hand side of FIG. 12A, modecoupler 1216 can be implemented using mode coupler 1216 (as describedabove), with a preceding and following phase shift (denoted by opensquares 1218 a, 1218 b). Thus, the transfer matrix T_(r) can beimplemented by the physical beam splitter shown in FIG. 12B, where theopen triangles represent +i phase shifters.

5.3. Example Photonic Spreading Circuits

Networks of mode couplers and phase shifters can be used to implementcouplings among more than two modes. For example, FIG. 13 shows afour-mode coupling scheme that implements a “spreader,” or“mode-information erasure,” transformation on four modes, i.e., it takesa photon in any one of the input modes and delocalizes the photonamongst each of the four output modes such that the photon has equalprobability of being detected in any one of the four output modes. (Thewell-known Hadamard transformation is one example of a spreadertransformation.) As in FIG. 11A, the horizontal lines 1312-1315correspond to modes, and the mode coupling is indicated by a verticalline 1316 with nodes (dots) to identify the modes being coupled. In thiscase, four modes are coupled. Circuit notation 1302 is an equivalentrepresentation to circuit diagram 1304, which is a network offirst-order mode couplings. More generally, where a higher-order modecoupling can be implemented as a network of first-order mode couplings,a circuit notation similar to notation 1302 (with an appropriate numberof modes) may be used.

FIG. 14 illustrates an example optical device 1400 that can implementthe four-mode mode-spreading transform shown schematically in FIG. 13 inaccordance with some embodiments. Optical device 1400 includes a firstset of optical waveguides 1401, 1403 formed in a first layer of material(represented by solid lines in FIG. 14) and a second set of opticalwaveguides 1405, 1407 formed in a second layer of material that isdistinct and separate from the first layer of material (represented bydashed lines in FIG. 14). The second layer of material and the firstlayer of material are located at different heights on a substrate. Oneof ordinary skill will appreciate that an interferometer such as thatshown in FIG. 14 could be implemented in a single layer if appropriatelow loss waveguide crossing were employed.

At least one optical waveguide 1401, 1403 of the first set of opticalwaveguides is coupled with an optical waveguide 1405, 1407 of the secondset of optical waveguides with any type of suitable optical coupler. Forexample, the optical device shown in FIG. 14 includes four opticalcouplers 1418, 1420, 1422, and 1424. Each optical coupler can have acoupling region in which two waveguides propagate in parallel. Althoughthe two waveguides are illustrated in FIG. 14 as being offset from eachother in the coupling region, the two waveguides may be positioneddirectly above and below each other in the coupling region withoutoffset. In some embodiments, one or more of the optical couplers 1418,1420, 1422, and 1424 are configured to have a coupling efficiency ofapproximately 50% between the two waveguides (e.g., a couplingefficiency between 49% and 51%, a coupling efficiency between 49.9% and50.1%, a coupling efficiency between 49.99% and 50.01%, and a couplingefficiency of 50%, etc.). For example, the length of the two waveguides,the refractive indices of the two waveguides, the widths and heights ofthe two waveguides, the refractive index of the material located betweentwo waveguides, and the distance between the two waveguides are selectedto provide the coupling efficiency of 50% between the two waveguides.This allows the optical coupler to operate like a 50/50 beam splitter.

In addition, the optical device shown in FIG. 14 can include twointer-layer optical couplers 1414 and 1416. Optical coupler 1414 allowstransfer of light propagating in a waveguide on the first layer ofmaterial to a waveguide on the second layer of material, and opticalcoupler 1416 allows transfer of light propagating in a waveguide on thesecond layer of material to a waveguide on the first layer of material.The optical couplers 1414 and 1416 allow optical waveguides located inat least two different layers to be used in a multi-channel opticalcoupler, which, in turn, enables a compact multi-channel opticalcoupler.

Furthermore, the optical device shown in FIG. 14 includes a non-couplingwaveguide crossing region 1426. In some implementations, the twowaveguides (1403 and 1405 in this example) cross each other withouthaving a parallel coupling region present at the crossing in thenon-coupling waveguide crossing region 1426 (e.g., the waveguides can betwo straight waveguides that cross each other at a nearly 90-degreeangle).

Those skilled in the art will understand that the foregoing examples areillustrative and that photonic circuits using beam splitters and/orphase shifters can be used to implement many different transfermatrices, including transfer matrices for real and imaginary Hadamardtransforms of any order, discrete Fourier transforms, and the like. Oneclass of photonic circuits, referred to herein as “spreader” or“mode-information erasure (MIE)” circuits, has the property that if theinput is a single photon localized in one input mode, the circuitdelocalizes the photon amongst each of a number of output modes suchthat the photon has equal probability of being detected in any one ofthe output modes. Examples of spreader or MIE circuits include circuitsimplementing Hadamard transfer matrices. (It is to be understood thatspreader or MIE circuits may receive an input that is not a singlephoton localized in one input mode, and the behavior of the circuit insuch cases depends on the particular transfer matrix implemented.) Inother instances, photonic circuits can implement other transfermatrices, including transfer matrices that, for a single photon in oneinput mode, provide unequal probability of detecting the photon indifferent output modes.

5.4. Example Photonic Bell State Generator Circuit

A Bell pair is a pair of qubits in any type of maximally entangled statereferred to as a Bell state. For dual rail encoded qubits, examples ofBell states (also referred to as the Bell basis states) include:

$ { { { | \Phi^{+}  \rangle = {\frac{  {  | 0  \rangle_{L} \middle| 0 \rangle_{L} +} \middle| 1 \rangle_{L} \middle| 1 \rangle_{L}}{\sqrt{2}} =  \frac{ { | {1010}  \rangle +} \middle| {0101} \rangle}{\sqrt{2}} \middle| \Phi^{-} }} \rangle = {\frac{  {  | 0  \rangle_{L} \middle| 0 \rangle_{L} -} \middle| 1 \rangle_{L} \middle| 1 \rangle_{L}}{\sqrt{2}} =  \frac{ { | {1010}  \rangle -} \middle| {0101} \rangle}{\sqrt{2}} \middle| \Psi^{+} }} \rangle = {\frac{  {  | 0  \rangle_{L} \middle| 1 \rangle_{L} +} \middle| 1 \rangle_{L} \middle| 0 \rangle_{L}}{\sqrt{2}} =  \frac{ { | {1001}  \rangle +} \middle| {0110} \rangle}{\sqrt{2}} \middle| \Psi^{-} }} \rangle = {\frac{  {  | 0  \rangle_{L} \middle| 1 \rangle_{L} -} \middle| 1 \rangle_{L} \middle| 0 \rangle_{L}}{\sqrt{2}} = \frac{ { | {1001}  \rangle -} \middle| {0110} \rangle}{\sqrt{2}}}$

In a computational basis (e.g., logical basis) with two states, aGreenberger-Horne-Zeilinger state is a quantum superposition of allqubits being in a first state of the two states superposed with all ofqubits being in a second state. Using logical basis described above, thegeneral M-qubit GHZ state can be written as:

$ | {GHZ}  \rangle = \frac{ { | 0  \rangle^{\otimes M} +} \middle| 1 \rangle^{\otimes M}}{\sqrt{2}}$

In some embodiments, entangled states of multiple photonic qubits can becreated by coupling modes of two (or more) qubits and performingmeasurements on other modes. By way of example, FIG. 15 shows a circuitdiagram for a Bell state generator 1500 that can be used in somedual-rail-encoded photonic embodiments. In this example, modes1532(1)-1532(4) are initially each occupied by a photon (indicated by awavy line); modes 1532(5)-1532(8) are initially vacuum modes. (Thoseskilled in the art will appreciate that other combinations of occupiedand unoccupied modes can be used.)

A first-order mode coupling (e.g., implementing transfer matrix T of Eq.(4)) is performed on pairs of occupied and unoccupied modes as shown bymode couplers 1531(1)-1531(4). Thereafter, a mode-information erasurecoupling (e.g., implementing a four-mode mode spreading transform asshown in FIG. 13) is performed on four of the modes (modes1532(5)-1532(8)), as shown by mode coupler 1537. Modes 1532(5)-1532(8)act as “heralding” modes that are measured and used to determine whethera Bell state was successfully generated on the other four modes1532(1)-1532(4). For instance, detectors 1538(1)-1538(4) can be coupledto the modes 1532(5)-1532(8) after second-order mode coupler 1537. Eachdetector 1538(1)-1538(4) can output a classical data signal (e.g., avoltage level on a conductor) indicating whether it detected a photon(or the number of photons detected). These outputs can be coupled toclassical decision logic circuit 1540, which determines whether a Bellstate is present on the other four modes 1532(1)-1532(4) based on theclassical output data. For example, decision logic circuit 1540 can beconfigured such that a Bell state is confirmed (also referred to as“success” of the Bell state generator) if and only if a single photonwas detected by each of exactly two of detectors 1538(1)-1538(4). Modes1532(1)-1532(4) can be mapped to the logical states of two qubits (Qubit1 and Qubit 2), as indicated in FIG. 15. Specifically, in this example,the logical state of Qubit 1 is based on occupancy of modes 1532(1) and1532(2), and the logical state of Qubit 2 is based on occupancy of modes1532(3) and 1532(4). It should be noted that the operation of Bell stategenerator 1500 can be non-deterministic; that is, inputting four photonsas shown does not guarantee that a Bell state will be created on modes1532(1)-1532(4). In one implementation, the probability of success is4/32.

In some embodiments, it is desirable to form resource states of multipleentangled qubits (typically 3 or more qubits, although the Bell statecan be understood as a resource state of two qubits). One technique forforming larger entangled systems is through the use of a “fusion” gate.A fusion gate receives two input qubits, each of which is typically partof an entangled system. The fusion gate performs a “fusion” operation onthe input qubits that produces either one (“type I fusion”) or zero(“type II fusion”) output qubits in a manner such that the initial twoentangled systems are fused into a single entangled system. Fusion gatesare specific examples of a general class of two-particle projectivemeasurements that can be employed to create entanglement between qubitsand are particularly suited for photonic architectures. Examples of typeI and type II fusion gates will now be described.

6. EXAMPLES OF FUSION GATE PHOTONIC CIRCUITS

FIGS. 16-21 show some embodiments of photonic circuit implementation offusion gates, or fusion circuits, for photonic qubits that can be usedaccording to some embodiments using Type II fusion. It should beunderstood that these example embodiments are illustrative and notlimiting. More generally, as used herein, the term “fusion gate” refersa device that can implement a two-particle projective measurement, e.g.,a Bell projection which, depending on the Bell basis chosen, can measuretwo operators e.g., the operators XX, ZZ, the operators XX, ZY, and thelike. A Type II fusion circuit (or gate), in the polarization encoding,takes two input modes, mixes them at a polarization beam splitter (PBS)and then rotates each of them by 45□ before measuring them in thecomputational basis. FIG. 16 shows an example. In the path encoding, aType II fusion circuit takes four modes, swaps the second and fourth,applies a 50:50 beamsplitter between the two pairs of adjacent modes andthen detects them all. FIG. 17 shows an example.

Fusion gates can be used in the construction of larger entangled statesby making use of the so-called “redundant encoding: of qubits. Thisconsists in a single qubit being represented by multiple photons, i.e.

α|0

+β|1

→α|0

^(⊗n)+β|0

^(⊗n),

so that the logical qubit is encoded in n individual qubits. This isachieved by measuring adjacent qubits in the X basis.

This encoding, denoted graphically as n qubits with no edges betweenthem (as in diagram (b) of FIG. 18), has the advantage that a Paulimeasurement on the redundant qubits does not split the cluster, butrather removes the photon measured from the redundant encoding andcombine the adjacent qubits into one single qubit that inherits thebonds of the input qubits, maybe adding a phase. In addition, anotheradvantage of this type of fusion is that it is loss tolerant. Both modesare measured, so there is no way to obtain the detection patterns thatherald success if one of the photons is lost. Finally, Type II fusiondoes not require the discrimination between different photon numbers, astwo detectors need to click for the heralding of successful fusion andthis can only happen if the photon count at each detector is 1.

The fusion succeeds with probability 50%, when a single photon isdetected at each detector in the polarization encoding. In this case, iteffectively performs a Bell state measurement on the qubits that aresent through it, projecting the pair of logical qubits into a maximallyentangled state. When the gate fails (as heralded by zero or two photonsat one of the detectors), it performs a measurement in the computationalbasis on each of the photons, removing them from the redundant encoding,but not destroying the logical qubit. The effect of the fusion in thegeneration of the cluster is depicted in FIGS. 18A-18D, where FIGS. 18Aand 18B show the measurement of a qubit in the linear cluster in the Xbasis to join it with its neighbor into a single logical qubit, andFIGS. 18C and 18D show the effect that success and failure of the gatehave on the structure of the cluster. It can be seen that a successfulfusion allows to build two-dimensional clusters.

A correspondence can be retrieved between the detection patterns and theKraus operators implemented by the gate on the state. In this case,since both qubits are detected, these are the projectors:

${h_{1}h_{2}}, {v_{1}v_{2}}arrow\frac{{h_{1}h_{\underset{¯}{2}}} + v_{12}}{\sqrt{2}} $${h_{1}v_{2}}, {v_{1}h_{2}}arrow\frac{{h_{1}h_{2}} - {v_{1}v_{2}}}{\sqrt{2}} $h₁², v₁² → ±h₁v₂ h₂², v₂² → ±v₁h₂,

where the first two lines correspond to ‘success’ outcomes, projectingthe two qubits into a Bell state, and the bottom two to ‘failure’outcomes, in which case the two qubits are projected into a productstate.

In some embodiments, the success probability of Type II fusion can beincreased by using ancillary Bell pairs or pairs of single photons.Employing a single ancilla Bell pair or two pairs of single photonsallows to boost the success probability to 75%.

One technique used to boost the fusion gate comes from the realizationthat, when it succeeds, it is equivalent to a Bell state measurement onthe input qubits. Therefore, increasing the success probability of thefusion gate corresponds to increasing that of the Bell state measurementit implements. Two different techniques to improve the probability ofdiscriminating Bell states have been developed by Grice (using a Bellpair) and Ewert & van Loock (https://arxiv.org/pdf/1403.4841.pdf) (usingsingle photons).

The former showed that an ancillary Bell pair allows to achieve asuccess probability of 75%, and the procedure can be iterated, usingincreasingly complex interferometers and large entangled states, toreach arbitrary success probability, in theory. However, the complexityof the circuit and the size of the entangled states necessary may makethis impractical.

The second technique makes use of four single photons, input in twomodes in pairs with opposite polarization, to boost the probability ofsuccess to 75%. It has also been shown numerically that the procedurecan be iterated a second time to obtain a probability of 78.125%, but ithas not been shown to be able to increase the success rate arbitrarilyas the other scheme.

FIG. 19 shows the Type II fusion gate boosted once using these twotechniques, both in polarization and path encoding. The successprobability of both circuits is 75%.

The detection patterns that herald success of the fusion are describedbelow for the two types of circuit.

When a Bell state is used to boost the fusion, the logic behind the‘success’ detection patterns is best understood by considering thedetectors in two pairs: the group corresponding to the input photonmodes (modes 1 and 2 in polarization and the top 4 modes inpath-encoding) and that corresponding to the Bell pair input modes(modes 3 and 4 in polarization and the bottom 4 modes in path-encoding).Call these the ‘main’ and ‘ancilla’ pairs respectively. Then asuccessful fusion is heralded whenever: (a) 4 photons are detected intotal; and (b) fewer than 4 photons are detected in each group ofdetectors.

When 4 single photons are used as ancillary resources, success of thegate is heralded whenever: (a) 6 photons are detected overall; and (b)fewer than 4 photons are detected at each detector.

When the gates succeeds, the two input qubits are projected onto one ofthe four Bell pairs, as these can be all discriminated from each otherthanks to the use of the ancillary resources. The specific projectiondepends on the detection pattern obtained, as before.

Both the boosted Type II fusion circuits, designed to take one Bell pairand four single photons as ancillae respectively, can be used to performType II fusion with variable success probabilities, if the ancillae arenot present or if only some of them are (in the case of the four singlephoton ancillae). This is particularly useful because it allows toemploy the same circuits to perform fusion in a flexible way, dependingon the resources available. If the ancillae are present, they can beinput in the gates to boost the probability of success of the fusion. Ifthey are not, however, the gates can still be used to attempt fusionwith a lower but non-zero success probability.

As far as the fusion gate boosted using one Bell pair is concerned, theonly case to be considered is that of the ancilla being absent. In thiscase, the logic of the detection patterns heralding success can beunderstood by considering the detectors in the pairs described aboveagain. The fusion is still successful when: (a) 2 photons are detectedat different detectors; and (b) 1 photon is detected in the ‘principal’pair and 1 photon is detected in the ‘ancilla’ pair of detectors.

In the case of the circuit boosted using four single photons, multiplemodifications are possible, removing all or part of the ancillae. Thisis analogous to the Boosted Bell State Generator, which is based on thesame principle.

First consider the case of no ancillae being present at all. Asexpected, the fusion is successful with probability 50%, which is thesuccess rate of the non-boosted fusion. In this case, the fusion issuccessful whenever 2 photons are detected at any two distinctdetectors.

As for the boosted BSG, the presence of an odd number of ancillae turnsout to be detrimental to the success probability of the gate: if 1photon is present, the gate only succeeds 32.5% of the time, whereas if3 photons are present, the success probability is 50%, like thenon-boosted case.

If only two of the four ancillae are present, two effects are possible.

If they are input in different modes in the polarization encoding, i.e.different adjacent pairs of ancillary modes in the path encoding, theprobability of success is lowered to 25%.

However, if the two ancillae are input in the same polarization mode,i.e. in the same pair of adjacent modes in the path encoding, thesuccess probability is boosted up to 62.5%. In this case, the patternsthat herald success can be understood again by grouping the detectors intwo pairs: the pair in the branch of the circuit where the ancillae areinput (group 1) and the pair in the other branch (group 2). Thisdistinction is particularly clear in the polarization-encoded diagram.Considering these groups, the fusion if successful when: (a) 4 photonsare detected overall; (b) fewer than 4 photons are detected at eachdetector in group 1; and (c) fewer than 2 photons are detected at eachdetector in group 2.

In these examples, the fusion gates work by projecting the input qubitsinto a maximally entangled state when successful. The basis such a stateis encoded in can be changed by introducing local rotations of the inputqubits before they enter the gate, i.e. before they are mixed at the PBSin the polarization encoding. Changing the polarization rotation of thephotons before they interfere at the PBS yields different subspaces ontowhich the state of the photons is projected, resulting in differentfusion operations on the cluster states. In the path encoding, thiscorresponds to applying local beamsplitters or combinations ofbeamsplitters and phase shifts corresponding to the desired rotationbetween the pairs of modes that constitute a qubit (neighboring pairs inthe diagrams above).

This can be useful to implement different types of cluster operations,both in the success and the failure cases, which can be very useful tooptimize the construction of a big cluster state from small entangledstates.

FIG. 20 shows a table with the effects of a few rotated variations ofthe Type II fusion gate used to fuse two small entangled states. Thediagram of the gate in the polarization encoding, the effectiveprojection performed and the final effect on the cluster state areshown.

Rotation to different basis states is further illustrated in FIG. 21,which shows examples of photonic circuits for Type II fusion gateimplementations using a path encoding. Shown are fusion gates for ZXfusion, XX fusion, ZZ fusion, and XZ fusion. In each instance acombination of beam splitters and phase shifters (e.g., as describedabove) can be used.

7. ADDITIONAL EMBODIMENTS

Those skilled in the art with access to this disclosure will appreciatethat embodiments described herein are illustrative and not limiting andthat many modifications and variations are possible. The measurementsperformed and the states on which they act can be chosen such that themeasurement outcomes have redundancies that give rise to faulttolerance. For instance, a code can be directly entered with themeasurements, or correlations can be generated in the measurements thatdirectly deal with both the destructiveness of the measurement and theentanglement breaking nature of the measurement in a fault tolerantmanner. This can be handled as part of the classical decoding; forinstance, failed fusion operations can be dealt with as erasure by thecode.

With reference to the appended figures, components that can includememory can include non-transitory machine-readable media. The terms“machine-readable medium” and “computer-readable medium” as used hereinrefer to any storage medium that participates in providing data thatcauses a machine to operate in a specific fashion. In embodimentsprovided hereinabove, various machine-readable media might be involvedin providing instructions/code to processors and/or other device(s) forexecution. Additionally or alternatively, the machine-readable mediamight be used to store and/or carry such instructions/code. In manyimplementations, a computer-readable medium is a physical and/ortangible storage medium. Such a medium may take many forms, including,but not limited to, non-volatile media, volatile media, and transmissionmedia. Common forms of computer-readable media include, for example,magnetic and/or optical media, punch cards, paper tape, any otherphysical medium with patterns of holes, a RAM, a programmable read-onlymemory (PROM), an erasable programmable read-only memory (EPROM), aFLASH-EPROM, any other memory chip or cartridge, a carrier wave asdescribed hereinafter, or any other medium from which a computer canread instructions and/or code.

The methods, systems, and devices discussed herein are examples. Variousembodiments may omit, substitute, or add various procedures orcomponents as appropriate. For instance, features described with respectto certain embodiments may be combined in various other embodiments.Different aspects and elements of the embodiments may be combined in asimilar manner. The various components of the figures provided hereincan be embodied in hardware and/or software. Also, technology evolvesand, thus, many of the elements are examples that do not limit the scopeof the disclosure to those specific examples.

It has proven convenient at times, principally for reasons of commonusage, to refer to such signals as bits, information, values, elements,symbols, characters, variables, terms, numbers, numerals, or the like.It should be understood, however, that all of these or similar terms areto be associated with appropriate physical quantities and are merelyconvenient labels. Unless specifically stated otherwise, as is apparentfrom the discussion above, it is appreciated that throughout thisspecification discussions utilizing terms such as “processing,”“computing,” “calculating,” “determining,” “ascertaining,”“identifying,” “associating,” “measuring,” “performing,” or the likerefer to actions or processes of a specific apparatus, such as a specialpurpose computer or a similar special purpose electronic computingdevice. In the context of this specification, therefore, a specialpurpose computer or a similar special purpose electronic computingdevice is capable of manipulating or transforming signals, typicallyrepresented as physical electronic, electrical, or magnetic quantitieswithin memories, registers, or other information storage devices,transmission devices, or display devices of the special purpose computeror similar special purpose electronic computing device.

Those of skill in the art will appreciate that information and signalsused to communicate the messages described herein may be representedusing any of a variety of different technologies and techniques. Forexample, data, instructions, commands, information, signals, bits,symbols, and chips that may be referenced throughout the abovedescription may be represented by voltages, currents, electromagneticwaves, magnetic fields or particles, optical fields or particles, or anycombination thereof.

Terms “and,” “or,” and “an/or,” as used herein, may include a variety ofmeanings that also is expected to depend at least in part upon thecontext in which such terms are used. Typically, “or” if used toassociate a list, such as A, B, or C, is intended to mean A, B, and C,here used in the inclusive sense, as well as A, B, or C, here used inthe exclusive sense. In addition, the term “one or more” as used hereinmay be used to describe any feature, structure, or characteristic in thesingular or may be used to describe some combination of features,structures, or characteristics. However, it should be noted that this ismerely an illustrative example and claimed subject matter is not limitedto this example. Furthermore, the term “at least one of” if used toassociate a list, such as A, B, or C, can be interpreted to mean anycombination of A, B, and/or C, such as A, B, C, AB, AC, BC, AA, AAB,ABC, AABBCCC, etc.

Reference throughout this specification to “one example,” “an example,”“certain examples,” or “exemplary implementation” means that aparticular feature, structure, or characteristic described in connectionwith the feature and/or example may be included in at least one featureand/or example of claimed subject matter. Thus, the appearances of thephrase “in one example,” “an example,” “in certain examples,” “incertain implementations,” or other like phrases in various placesthroughout this specification are not necessarily all referring to thesame feature, example, and/or limitation. Furthermore, the particularfeatures, structures, or characteristics may be combined in one or moreexamples and/or features.

In some implementations, operations or processing may involve physicalmanipulation of physical quantities. Typically, although notnecessarily, such quantities may take the form of electrical or magneticsignals capable of being stored, transferred, combined, compared, orotherwise manipulated. It has proven convenient at times, principallyfor reasons of common usage, to refer to such signals as bits, data,values, elements, symbols, characters, terms, numbers, numerals, or thelike. It should be understood, however, that all of these or similarterms are to be associated with appropriate physical quantities and aremerely convenient labels. Unless specifically stated otherwise, asapparent from the discussion herein, it is appreciated that throughoutthis specification discussions utilizing terms such as “processing,”“computing,” “calculating,” “determining,” or the like refer to actionsor processes of a specific apparatus, such as a special purposecomputer, special purpose computing apparatus or a similar specialpurpose electronic computing device. In the context of thisspecification, therefore, a special purpose computer or a similarspecial purpose electronic computing device is capable of manipulatingor transforming signals, typically represented as physical electronic ormagnetic quantities within memories, registers, or other informationstorage devices, transmission devices, or display devices of the specialpurpose computer or similar special purpose electronic computing device.

In the preceding detailed description, numerous specific details havebeen set forth to provide a thorough understanding of claimed subjectmatter. However, it will be understood by those skilled in the art thatclaimed subject matter may be practiced without these specific details.In other instances, methods and apparatuses that would be known by oneof ordinary skill have not been described in detail so as not to obscureclaimed subject matter. Therefore, it is intended that claimed subjectmatter not be limited to the particular examples disclosed, but thatsuch claimed subject matter may also include all aspects falling withinthe scope of appended claims, and equivalents thereof.

For an implementation involving firmware and/or software, themethodologies may be implemented with modules (e.g., procedures,functions, and so on) that perform the functions described herein. Anymachine-readable medium tangibly embodying instructions may be used inimplementing the methodologies described herein. For example, softwarecodes may be stored in a memory and executed by a processor unit. Memorymay be implemented within the processor unit or external to theprocessor unit. As used herein the term “memory” refers to any type oflong term, short term, volatile, nonvolatile, or other memory and is notto be limited to any particular type of memory or number of memories, ortype of media upon which memory is stored.

If implemented in firmware and/or software, the functions may be storedas one or more instructions or code on a computer-readable storagemedium. Examples include computer-readable media encoded with a datastructure and computer-readable media encoded with a computer program.Computer-readable media includes physical computer storage media. Astorage medium may be any available medium that can be accessed by acomputer. By way of example, and not limitation, such computer-readablemedia can comprise RAM, ROM, EEPROM, compact disc read-only memory(CD-ROM) or other optical disk storage, magnetic disk storage,semiconductor storage, or other storage devices, or any other mediumthat can be used to store desired program code in the form ofinstructions or data structures and that can be accessed by a computer;disk and disc, as used herein, includes compact disc (CD), laser disc,optical disc, digital versatile disc (DVD), floppy disk and Blu-ray discwhere disks usually reproduce data magnetically, while discs reproducedata optically with lasers. Combinations of the above should also beincluded within the scope of computer-readable media.

In addition to storage on computer-readable storage medium, instructionsand/or data may be provided as signals on transmission media included ina communication apparatus. For example, a communication apparatus mayinclude a transceiver having signals indicative of instructions anddata. The instructions and data are configured to cause one or moreprocessors to implement the functions outlined in the claims. That is,the communication apparatus includes transmission media with signalsindicative of information to perform disclosed functions. At a firsttime, the transmission media included in the communication apparatus mayinclude a first portion of the information to perform the disclosedfunctions, while at a second time the transmission media included in thecommunication apparatus may include a second portion of the informationto perform the disclosed functions.

All patent applications, patents, and printed publications cited hereinare incorporated herein by reference in their entireties, except for anydefinitions, subject matter disclaimers or disavowals, and except to theextent that the incorporated material is inconsistent with the expressdisclosure herein, in which case the language in this disclosurecontrols.

What is claimed is:
 1. A method comprising: receiving, by a qubit fusionsystem, a plurality of quantum systems, wherein each quantum system ofthe plurality of quantum system includes a plurality of quantumsub-systems in an entangled state, and wherein respective quantumsystems of the plurality of quantum systems are independent quantumsystems that are not entangled with one another; performing, by thequbit fusion system, a plurality of joint measurements on differentquantum sub-systems from respective ones of the plurality of quantumsystems, wherein the joint measurements generate joint measurementoutcome data; and determining, by a decoder, a plurality of syndromegraph values based on the joint measurement outcome data.
 2. The methodof claim 1, wherein performing the joint measurements includesperforming fusion operations.
 3. The method of claim 1, whereinperforming the joint measurements include performing a destructive jointmeasurement via a Type II fusion operation.
 4. The method of claim 1,wherein performing the plurality of joint measurements on differentquantum sub-systems from respective ones of the plurality of quantumsystems includes performing the plurality of joint measurements on onlya subset of the plurality of quantum sub-systems that are received bythe qubit fusion system thereby resulting in a subset of unmeasuredquantum sub-systems.
 5. The method of claim 4, further comprising,receiving, by the qubit fusion system, a second plurality of quantumsystems, wherein each quantum system of the second plurality of quantumsystem includes a second plurality of quantum sub-systems in anentangled state, and wherein respective quantum systems of the secondplurality of quantum systems are independent quantum systems that arenot entangled with one another; receiving the subset of unmeasuredquantum sub-systems; and performing, by the qubit fusion system, asecond plurality of joint measurements between i) second quantumsub-systems from respective ones of the plurality of second quantumsystems and ii) respective quantum sub-systems from the subset ofunmeasured quantum sub-systems, wherein the second plurality of jointmeasurements generate second joint measurement outcome data.
 6. A systemcomprising: a qubit fusion system comprising a plurality of fusiongates, wherein the qubit fusion system is configured to receive aplurality of quantum systems, wherein each quantum system of theplurality of quantum system includes a plurality of quantum sub-systemsin an entangled state, and wherein respective quantum systems of theplurality of quantum systems are independent quantum systems that arenot entangled with one another; wherein the plurality of fusion gatesare each configured to perform a joint measurement on different quantumsub-systems from respective ones of the plurality of quantum systems,wherein the joint measurements generate joint measurement outcome data;and a decoder communicatively coupled to the qubit fusion system andconfigured to receive the joint measurement outcome data and todetermine a plurality of syndrome graph values based on the jointmeasurement outcome data.
 7. The system of claim 6, wherein the fusiongates include a photonic circuit and the plurality of quantum systemscomprise photons as the quantum sub-systems, wherein the photoniccircuit comprises a Type II fusion gate.
 8. The system of claim 6,wherein the joint measurement comprises a two-particle projectivemeasurement onto a Bell basis.
 9. The system of claim 6, furthercomprising a quantum memory, coupled to at least one the qubit fusionsystem and that receive and store a subset of the plurality of quantumsub-systems.
 10. The system of claim 9, wherein the quantum memory is anoptical fiber.
 11. The system of claim 9, wherein the quantum memory iscoupled to the qubit fusion system such that the joint measurement isperformed between i) quantum sub-systems from respective ones of theplurality of quantum systems and ii) respective quantum sub-systems fromthe subset of the plurality of quantum sub-systems that are stored inthe quantum memory.
 12. The system of claim 6 further comprising a qubitentangling system that is configured to generate the plurality ofquantum systems.
 13. The system of claim 12, wherein the qubitentangling system includes a quantum gate array.
 14. The system of claim13, wherein qubit entangling system includes a photon source system thatis optically connected to an entangled state generator.
 15. The systemof claim 14, wherein the entangled state generator is configured toreceive output photons from the photon source system and convert theoutput photons to an entangled photonic state.
 16. The system of claim15, wherein the qubit entangling system includes a plurality of outputwaveguides that are optically coupled to the qubit fusion system and areconfigured to provide the entangled photonic state to inputs of thefusion gates.